Research Article Bayesian Inference for Step-Stress Partially Accelerated...
11
Research Article Bayesian Inference for Step-Stress Partially Accelerated Competing Failure Model under Type II Progressive Censoring Xiaolin Shi, 1 Fen Liu, 2 and Yimin Shi 2 1 School of Electronic Engineering, Xi’an University of Posts and Telecommunications, Xi’an 710121, China 2 Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, China Correspondence should be addressed to Xiaolin Shi; [email protected] Received 4 July 2016; Revised 4 November 2016; Accepted 17 November 2016 Academic Editor: Jos´ e A. Sanz-Herrera Copyright © 2016 Xiaolin Shi et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper deals with the Bayesian inference on step-stress partially accelerated life tests using Type II progressive censored data in the presence of competing failure causes. Suppose that the occurrence time of the failure cause follows Pareto distribution under use stress levels. Based on the tampered failure rate model, the objective Bayesian estimates, Bayesian estimates, and E- Bayesian estimates of the unknown parameters and acceleration factor are obtained under the squared loss function. To evaluate the performance of the obtained estimates, the average relative errors (AREs) and mean squared errors (MSEs) are calculated. In addition, the comparisons of the three estimates of unknown parameters and acceleration factor for different sample sizes and different progressive censoring schemes are conducted through Monte Carlo simulations. 1. Introduction e competing failure model is commonly used in medical studies or reliability analysis. Due to the complexity in the internal structure and the external working environment, there are oſten many causes of the product failure. Each of these causes is possible to result in final failure of the product. is is called competing failure. In recent years, some of authors have investigated the competing failure models. Han and Kundu [1] studied the statistical analysis of the step- stress accelerated life test (SSALT) with competing risks for failure from the generalized exponential distribution under Type I censoring, and the point estimation and confidence interval of the distribution parameters are obtained. In the case that failure follows the lognormal distribution, Roy and Mukhopadhyay [2] investigated the maximum likelihood analysis of the accelerated life test of competing failure products using the EM algorithm. Zhang et al. [3] applied the copula model to combine the marginal distribution of the competing failure models with the time to failure distribution of the products and derived a general statistical model for accelerated life test (ALT) with s-dependent competing failures. Shi et al. [4] considered a constant-stress accelerated life test (CSALT) with competing risks for failure from exponential distribution under progressive Type II hybrid censoring. e maximum likelihood estimator and Bayes estimator of the parameter were derived. Xu and Tang [5] and Wu et al. [6] considered different inferential issues regarding the constant-stress accelerated competing failure models when the lifetime of different risk factors follows Weibull distribution. Other research results can be found in [7, 8]. e above references considered competing failure mod- els in the case of CSALT or SSALT. Based on the CSALT or SSALT to analyze the life characteristics of product, we need to use the relationship between product life and stress levels, that is, acceleration model. But, in engineering practice, the accelerated model is not always known, such as the newly developed products. In order to solve this problem, the partially accelerated life test (PALT) is more suitable to be performed. ere are two types of PALT, namely, constant- stress PALT (CSPALT) and step-stress PALT (SSPALT). e latter allows for changing the stress from normal level to accelerated level at a specific time point, which contributes to getting the information of the products’ lifetimes quickly. Under the PALT, Abd-Elfattah et al. [9] and Ismail [10, 11] Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2016, Article ID 2097581, 10 pages http://dx.doi.org/10.1155/2016/2097581
Transcript of Research Article Bayesian Inference for Step-Stress Partially Accelerated...
Research ArticleBayesian Inference for Step-Stress Partially AcceleratedCompeting Failure Model under Type II Progressive Censoring
Xiaolin Shi1 Fen Liu2 and Yimin Shi2
1School of Electronic Engineering Xirsquoan University of Posts and Telecommunications Xirsquoan 710121 China2Department of Applied Mathematics Northwestern Polytechnical University Xirsquoan 710072 China
Correspondence should be addressed to Xiaolin Shi linda20016163com
Received 4 July 2016 Revised 4 November 2016 Accepted 17 November 2016
Academic Editor Jose A Sanz-Herrera
Copyright copy 2016 Xiaolin Shi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper deals with the Bayesian inference on step-stress partially accelerated life tests using Type II progressive censored datain the presence of competing failure causes Suppose that the occurrence time of the failure cause follows Pareto distributionunder use stress levels Based on the tampered failure rate model the objective Bayesian estimates Bayesian estimates and E-Bayesian estimates of the unknown parameters and acceleration factor are obtained under the squared loss function To evaluatethe performance of the obtained estimates the average relative errors (AREs) and mean squared errors (MSEs) are calculated Inaddition the comparisons of the three estimates of unknown parameters and acceleration factor for different sample sizes anddifferent progressive censoring schemes are conducted through Monte Carlo simulations
1 Introduction
The competing failure model is commonly used in medicalstudies or reliability analysis Due to the complexity in theinternal structure and the external working environmentthere are often many causes of the product failure Each ofthese causes is possible to result in final failure of the productThis is called competing failure In recent years some ofauthors have investigated the competing failure models Hanand Kundu [1] studied the statistical analysis of the step-stress accelerated life test (SSALT) with competing risks forfailure from the generalized exponential distribution underType I censoring and the point estimation and confidenceinterval of the distribution parameters are obtained In thecase that failure follows the lognormal distribution Roy andMukhopadhyay [2] investigated the maximum likelihoodanalysis of the accelerated life test of competing failureproducts using the EM algorithm Zhang et al [3] appliedthe copula model to combine themarginal distribution of thecompeting failuremodels with the time to failure distributionof the products and derived a general statistical modelfor accelerated life test (ALT) with s-dependent competingfailures Shi et al [4] considered a constant-stress accelerated
life test (CSALT) with competing risks for failure fromexponential distribution under progressive Type II hybridcensoring The maximum likelihood estimator and Bayesestimator of the parameter were derived Xu and Tang [5]and Wu et al [6] considered different inferential issuesregarding the constant-stress accelerated competing failuremodels when the lifetime of different risk factors followsWeibull distribution Other research results can be found in[7 8]
The above references considered competing failure mod-els in the case of CSALT or SSALT Based on the CSALT orSSALT to analyze the life characteristics of product we needto use the relationship between product life and stress levelsthat is acceleration model But in engineering practice theaccelerated model is not always known such as the newlydeveloped products In order to solve this problem thepartially accelerated life test (PALT) is more suitable to beperformed There are two types of PALT namely constant-stress PALT (CSPALT) and step-stress PALT (SSPALT) Thelatter allows for changing the stress from normal level toaccelerated level at a specific time point which contributesto getting the information of the productsrsquo lifetimes quicklyUnder the PALT Abd-Elfattah et al [9] and Ismail [10 11]
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 2097581 10 pageshttpdxdoiorg10115520162097581
2 Mathematical Problems in Engineering
derived the maximum likelihood estimates and Bayesianestimates of the acceleration factor and unknown parametersfor different lifetime distributions Abushal and Soliman [12]studied the MLE and Bayesian estimate of the parameters forthe Pareto distribution of the second type under progressivecensoring data for constant PALT Ismail and Sarhan [13]discussed optimal design of step-stress life test with progres-sively Type II censored exponential data But few authors havestudied competing failure models from this perspective andespecially from the Bayesian viewpoint
The Bayesian method has certain accurate advantageswhen sample size is small When little knowledge about theprior information is obtained we can derive the objectiveBayesian estimates based on noninformative priors such asthe Jeffreys prior and reference prior For more details seeJeffreys [14] Guan et al [15] derived the objective Bayesianestimates of unknown parameters for bivariate Marshall-Olkin exponential distribution respectively Under Type Icensoring Xu and Tang [5] considered the objective Bayesianinference for the accelerated competing failure modelsInstead with historical data or experience of some expertsavailable the subjective Bayesian approach on the basis ofinformative prior distributions can be implemented whileif the hyperparameters of the informative prior distributionsare unknownwe are able to use E-Bayesianmethod proposedby Han [16 17] By assuming the hyperparameters followcertain prior distributions the E-Bayesian estimate can beobtained as the expectation of the Bayesian estimate of theunknown parameter
In life testing and reliability studies the experimentermay not always obtain complete information on failure timesfor all experimental units Thus censoring is common tobe performed A censoring scheme (CS) which can balancebetween total time spent for the experiment number ofunits used in the experiment and the efficiency of statisticalinference based on the results of the experiment is desirable[12] Compared to the conventional Type I and Type IIcensoring scheme Type II progressive censoring schemeprovides higher flexibility to the experimenter by allowingfor product to be removed at nonterminal time points Thisallowance may be desirable when some degradation-relatedinformation from live test specimens needs to be collectedand analyzed or when scarce testing facilities need to bereleased for other experimentation In recent years some ofresearchers engaged in reliability research using progressivelycensored data such as Abushal and Soliman [12] and Ismailand Sarhan [13]
The Pareto distribution has found widespread use as amodel for various socioeconomic phenomena see [18 19]The Pareto distribution has also been used in reliability andlifetime modelling (see [20ndash22]) In this paper we will con-sider Bayesian analysis for the step-stress partially acceleratedcompeting failuremodel fromPareto distribution under TypeII progressive censoring The rest of this paper is organizedas follows In Section 2 under Type II progressive censoringscheme a step-stress partially accelerated competing failuremodel from Pareto distribution is described and some basicassumptions are given In Section 3 we obtain the objectiveBayesian estimates of the acceleration factor and unknown
parameters based on the noninformative prior distributionFurthermore the Bayesian and corresponding E-Bayesianestimates are provided In Section 4 we provide two real datasets to justify the use of the Pareto distribution in engineeringpractice The simulation results of all proposed methods fordifferent sample sizes and for different progressive censoringschemes are presented in Section 5 and some conclusions arepresented in Section 6
2 Assumptions and Model Description
Throughout the paper we make the following basic assump-tions
(1) The failure of a product occurs only due to one of the2 independent competing failure causes with lifetimes1198791 and 1198791 Then the lifetime of the product is 119879 =min(1198791 1198792)
(2) Two stress levels 1198780 and 1198781 are used in SSPALT (1198780 lt1198781) where 1198780 is use stress level and 1198781 is acceleratedstress level
(3) Let 1198791119895 be the occurrence time of the 119895th failure causeunder stress levels 1198780 and 1198792119895 be the occurrence timeof the 119895th failure cause under stress levels 1198781 119895 =1 2 The times 1198791119895 and 1198792119895 are independent Also1198791119895 follows a Pareto distribution with the followinghazard rate function (HRF) and reliability function
1199031119895 (119905) = 120572119895119905 119905 ge 1205791198951198781119895 (119905) = 120579120572119895119895 119905minus120572119895 119905 ge 120579119895 119895 = 1 2
(1)
where 120579119895 gt 0 is the scale parameter and 120572119895 gt 0 is theshape parameter 119895 = 1 2
(4) Tampered failure ratemodel holds that is theHRF 1199031119895is proportional to 1199032119895 Thus the HRF of the jth failurecause in SSPALT can be expressed as
119903lowast119895 (119905) =1199031119895 (119905) = 120572119895119905 120579119895 le 119905 le 1205911199032119895 (119905) = 120572119895120582119905 119905 gt 120591 120582 gt 1 (2)
where 120591 is prefixed number 120591 isin (0infin) and 120582 is accel-eration factor The corresponding reliability functionis
119878lowast119895 (119905) = 1198781119895 (119905) = 120579120572119895119895 119905minus120572119895 120579119895 le 119905 le 1205911198782119895 (119905) = 120579120572119895119895 120591120572119895(120582minus1)119905minus120572119895120582 119905 gt 120591 120582 gt 1 (3)
Under Type II progressive censored scheme SSPALT isdescribed as follows 119899 identical products are placed on thetest under the stress level 1198780 At the time of the first failure1198771 products are randomly removed from the remaining 119899 minus 1products and the failure cause index 1198881 is recorded Similarlyat the time of the second failure 1198772 products are randomly
Mathematical Problems in Engineering 3
removed from the remaining 119899 minus 2 minus 1198771 products and thefailure cause index 1198882 is recordedThe test continues until timepoint 120591 the stress levels are changed from 1198780 to 1198781 Under thestress level 1198781 the remaining 119899 minus 1198731 minus 1198771 minus 1198772 minus sdot sdot sdot minus 1198771198731products continue to be tested Suppose that the total numberof failures under the stress level 1198780 is 1198731 At the time of the(1198731 + 1)th failure 1198771198731+1 products are randomly removedfrom the remaining 119899 minus (1198731 + 1) minus 1198771 minus 1198772 minus sdot sdot sdot minus 1198771198731products and the failure cause index 1198881198731+1 is recorded Thetest continues until themth failure is observed and the failurecause index 119888119898 is recorded At this time all the remaining119877119898 = 119899minus119898minus1198771minus1198772minussdot sdot sdotminus119877119898minus1 products are removed and thetest terminates where119898 119877119894 (119894 = 1 2 119898 minus 1) are prefixednumber (1198731 lt 119898 lt 119899) and 119888119894 isin 1 2 119894 = 1 2 119898 Wedenote the time of the ith failure by 119905119894 and assume that eachfailure time alongwith the failure cause index is observed andthere is at least one failure caused by the jth (119895 = 1 2) failurecause under stress level 119878119896 (119896 = 0 1) Therefore observeddata can be denoted as (1199051 1198881) (1199052 1198882) (119905119898 119888119898) Then thelikelihood function of the failure samples caused by the 119895thfailure cause is
119871119895prop 1198731prod119894=1
1199031119895120575119895(119888119894) (119905119894) 1198781+1198771198941119895 (119905119894) 119898prod119894=1198731+1
1199032119895120575119895(119888119894) (119905119894) 1198781+1198771198942119895 (119905119894)= 120572119895119899119895120579120572119895119899119895 120582119899211989511988011198951198802119895119882minus1205721198951205821 119882120572119895(120582minus1)2
(4)
where
120575119895 (119888119894) = 1 119888119894 = 1198950 119888119894 = 119895
1198801119895 = 1198731prod119894=1
119905minus120575119895(119888119894)minus120572119895(1+119877119894)119894 1198802119895 = 119898prod
119894=1198731+1
119905minus120575119895(119888119894)119894 1198821 = 119898prod
119894=1198731+1
119905(1+119877119894)119894 1198822 = 120591119860(119898)119899119895 = 1198991119895 + 11989921198951198991119895 = 1198731sum
119894=1
120575119895 (119888119894) 1198992119895 = 119898sum
119894=1198731+1
120575119895 (119888119894) 119860 (119898) = 119898sum
119894=1198731+1
(1 + 119877119894)
(5)
Based on assumptions (1) and (4) we derive the full-likelihood function as
119871 (119905 | 120582 1205791 1205792) = 2prod119895=1
119871119895 prop ( 2prod119895=1
120572119899119895119895 11988011198951198802119895120579120572119895119899119895 )sdot 120582119898minus1198731119882minus(1205721+1205722)1205821 119882(1205721+1205722)(120582minus1)2
(6)
3 Bayesian Analysis
In this section the Bayesian method is employed to estimatethe scale parameters and acceleration factor when the shapeparameters are known Depending on whether the priorinformation is available or not we obtain the objectiveBayesian estimates Bayesian estimates and E-Bayesian esti-mates of the unknown parameters and acceleration factorunder the squared loss function respectively
31 Objective Bayesian Estimates When little knowledgeabout the prior information is obtained the objectiveBayesian approach is an alternative way We assume that thejoint prior distribution of the parameters (120582 1205791 1205792) is
120587119900 (120582 1205791 1205792) prop 112058212057911205792 (7)
From (6) and (7) the posterior density functions of theparameters 120582 1205791 and 1205792 can be given by 120587119900(120582 1205791 1205792 | 119905) prop119871(119905 | 120582 1205791 1205792)120587119900(120582 1205791 1205792) Then the marginal posteriordensity functions of the parameters 120582 1205791 and 1205792 can beobtained respectively
120587119900 (120582 | 119905) = 1198631120582119898minus1198731minus1119882minus(1205721+1205722)1205821 119882(1205721+1205722)1205822 120582 gt 1120587119900 (1205791 | 119905) = 11986321205791205721119899minus11 0 lt 1205791 lt 11990511120587119900 (1205792 | 119905) = 11986331205791205722119899minus12 0 lt 1205792 lt 11990521
(8)
where 1199051198961 is the first-failure time caused by failure cause119896 (119896 = 1 2) and1198631 = [intinfin1 120582119898minus1198731minus1119882minus(1205721+1205722)1205821 119882(1205721+1205722)1205822 119889120582]minus11198632 = [int119905110 1205791205721119899minus11 1198891205791]minus1 and1198633 = [int119905210 1205791205721119899minus12 1198891205792]minus1
Under the squared loss function the objective Bayesianestimates of the parameters 120582 1205791 and 1205792 can be obtained as
OBS = 1198631 intinfin1120582119898minus1198731119882minus(1205721+1205722)1205821 119882(1205721+1205722)1205822 119889120582
119895OBS = 1205721198951198991199051198951 (120572119895119899 + 1)minus1 119895 = 1 2(9)
It is obvious to see that (9) does not have explicit form Sowe obtain the Bayesian estimate of 120582 by numerical methodLet 119891(120582) = 120582119898minus1198731minus1119882minus(1205721+1205722)1205821 119882(1205721+1205722)1205822 119881 = 120582(1205721 +1205722) log(11988211198822) and then
intinfin1120582119898minus1198731minus1119882minus(1205721+1205722)1205821 119882(1205721+1205722)1205822 119889120582 = intinfin
0119891 (120582) 119889120582
minus int10119891 (120582) 119889120582 = [(1205721 + 1205722) log(11988211198822)]
minus(119898minus1198731)
4 Mathematical Problems in Engineering
sdot [intinfin0119881119898minus1198731minus1119890minus119881119889119881 minus int119881lowast
0119881119898minus1198731minus1119890minus119881119889119881]
= [(1205721 + 1205722) log(11988211198822)]minus(119898minus1198731)
sdot [Γ (119898 minus 1198731) minus int119881lowast
0119881119898minus1198731minus1119890minus119881119889119881]
(10)
where 119881lowast = (1205721 + 1205722) log(11988211198822) Similarly we get
intinfin1120582119891 (120582) 119889120582 = intinfin
0120582119891 (120582) 119889120582 minus int1
0120582119891 (120582) 119889120582
= [(1205721 + 1205722) log(11988211198822)]minus(119898minus1198731+1)
sdot [Γ (119898 minus 1198731 + 1) minus int119881lowast
0119881119898minus1198731119890minus119881119889119881]
(11)
Thus under the squared loss function the objective Bayesianestimates of the parameters 120582 can be obtained as
OBS = 1198631 intinfin1120582119898minus1198731119882minus(1205721+1205722)1205821 119882(1205721+1205722)1205822 119889120582 = intinfin1 120582119891 (120582) 119889120582intinfin
1119891 (120582) 119889120582
= [(1205721 + 1205722) log (11988211198822)] [Γ (119898 minus 1198731 + 1) minus int119881lowast
0119881119898minus1198731119890minus119881119889119881]
[Γ (119898 minus 1198731) minus int119881lowast0 119881119898minus1198731minus1119890minus119881119889119881] (12)
About the integralsint119881lowast0119881119898minus1198731119890minus119881119889119881 andint119881lowast
0119881119898minus1198731minus1119890minus119881119889119881
the compound trapezoidal integral formula can be used toobtain integral values The specific steps are as follows theinterval [0 119881lowast] is divided into 119899 equal subintervals and eachlength of subinterval is ℎ = 119881lowast119899 Then from the compoundtrapezoidal integral formula we get
int119881lowast0119892 (119881) 119889119881asymp ℎ2 [119892 (0) + 2
119899minus1sum119896=1
119892(119881lowast119896119899 ) + 119892 (119881lowast)] int119881lowast0119891 (119881) 119889119881asymp ℎ2 [119891 (0) + 2
119899minus1sum119896=1
119891(119881lowast119896119899 ) + 119891 (119881lowast)]
(13)
where 119892(119881) = 119881119898minus1198731119890minus119881 and 119891(119881) = 119881119898minus1198731minus1119890minus119881 Theabove two integration results are substituted into (12) andthe objective Bayesian estimates of the parameters 120582 can beobtained
32 Bayesian Estimates With historical data or expertsrsquo expe-rience available the informative prior could be a betteroption Following Doostparast et al [23] a convenient priorfamily of distributions for 120579119895 (119895 = 1 2) is provided bythe power function distributions with probability densityfunction
120587119904 (120579119895) = 120573119895120579120573119895minus1119895 119889minus120573119895119895 0 lt 120579119895 lt 119889119895 119889119895 120573119895 gt 0 (14)
We consider the prior density function of 120582 as follows120587119904 (120582) = 119860120582119886minus1 exp minus119887120582 119886 119887 gt 0 120582 gt 1 (15)
where 119860 is the regularization factor
Then the marginal posterior density functions of theparameters 120582 1205791 and 1205792 are respectively120587119904 (120582 | 119905) prop 120582119898minus1198731+119886minus1sdot exp minus [(1205721 + 1205722) (ln1198821 minus ln1198822) + 119887] 120582
119886 119887 gt 0 120582 gt 1120587119904 (120579119895 | 119905) = (120572119895119899 + 120573119895) 120579
120572119895119899+120573119895minus1
119895
119889lowast(120572119895119899+120573119895)119895
0 lt 120579119895 lt 119889lowast119895 119889lowast119895 = min (1199051198951 119889119895)
(16)
where 1199051198951 is the first-failure time caused by the 119895th failurecause (119895 = 1 2)
Under the squared loss function the Bayesian estimate of120579119895 (119895 = 1 2) can be easily obtained as
119895BS = int119889lowast119895
0120579119895120587119878 (120579119895 | 119905) 119889120579119895 = (120572119895119899 + 120573119895) 119889lowast119895(120572119895119899 + 120573119895 + 1) (17)
To obtain the Bayesian estimate of 120582 we expand 119867(120582) =120582119898minus1198731+119886 and ℎ(120582) = 120582119898minus1198731+119886minus1 in Taylor series aroundthe maximum likelihood estimate to get the followingapproximation
119867(120582) asymp 119867() + 1198671015840 () (120582 minus ) ℎ (120582) asymp ℎ () + ℎ1015840 () (120582 minus ) (18)
where1198671015840(120582) = (119898minus1198731 +119886)120582119898minus1198731+119886minus1 and ℎ1015840(120582) = (119898minus1198731 +119886 minus 1)120582119898minus1198731+119886minus2
Mathematical Problems in Engineering 5
Therefore under the squared loss function the Bayesianestimate of 120582 can be approximated as
BS = 119867() + 1198671015840 () (1 + 119868minus11 minus )ℎ () + ℎ1015840 () (1 + 119868minus11 minus ) (19)
where 1198681 = (1205721 + 1205722)(ln1198821 minus ln1198822) + 11988733 E-Bayesian Estimates By assuming that the hyperparam-eters follow certain distributions the Bayesian estimates canbe obtained when the hyperparameters in prior distributionsare unknown However complicated integration is often ahard work Han [16 17] gave the definition of E-Bayesianestimation to overcome this problem So we will derivethe E-Bayesian estimates of the unknown parameters andacceleration factor in this subsection
Suppose that the hyperparameters 119886 and 119887 and 120573119895 and119889119895 (119895 = 1 2) have the joint prior density functions 120587119890(119886 119887)and 120587119890(120573119895 119889119895) respectively Then the E-Bayesian estimates of120582 and 120579119895 (119895 = 1 2) are
EB = ∬119863BS120587119890 (119886 119887) 119889119886 119889119887 (20)
119895EB = ∬119863119895BS120587119890 (120573119895 119889119895) 119889120573119895119889 (119889119895) (21)
where BS and 119895BS are the Bayesian estimates of 120582 and 120579119895 (119895 =1 2) respectivelyAccording to Han [16] 119886 and 119887 and 120573119895 and 119889119895 (119895 = 1 2)
should be selected to guarantee that 120587119904(120582) and 120587119904(120579119895) aredecreasing functions of 120582 and 120579119895 respectivelyThe derivativesof 120587119904(120582) with respect to 120582 and 120587119904(120579119895) with respect to 120579119895 are
120597120587119904 (120582)120597120582 prop 120582119886minus2 exp minus119887120582 (119886 minus 1 minus 119887120582) 120597120587119904 (120579119895)120597120579119895 = 120573119895 (120573119895 minus 1) 120579120573119895minus2119895 119889minus120573119895119895
(22)
When 0 lt 119886 lt 1 119887 gt 0 120597120587119904(120582)120597120582 lt 0 and when 0 lt 120573119895 lt 1119889119895 gt 0 120597120587119904(120579119895)120597120579119895 lt 0Thus120587119904(120582) and120587119904(120579119895) are decreasingfunctions of 120582 and 120579119895 respectively
Given that 0 lt 119886 lt 1 and 0 lt 120573119895 lt 1 thelarger 119887 and 119889119895 are the thinner the tail of 120587119904(120582) and 120587119904(120579119895)will be Berger [24] showed that the thinner tailed priordistribution often reduces the robustness of the Bayesianestimate Consequently 119887 and 119889119895 (119895 = 1 2) should begiven upper bounds 119897 and ℎ1015840119895 respectively We consider thefollowing two joint prior distributions of 119886 and 119887 and 120573119895 and119889119895 (119895 = 1 2) respectively
1205871 (119886 119887) = 21198871198972 0 lt 119886 lt 1 0 lt 119887 lt 1198971205872 (119886 119887) = 1119897 0 lt 119886 lt 1 0 lt 119887 lt 119897 (23)
1205871119895 (120573119895 119889119895) = 2119889119895ℎ10158402119895 0 lt 120573119895 lt 1 0 lt 119889119895 lt ℎ10158401198951205872119895 (120573119895 119889119895) = 1ℎ1015840119895 0 lt 120573119895 lt 1 0 lt 119889119895 lt ℎ1015840119895
(24)
From (19) (20) and (23) we derive the E-Bayesian estimatesof 120582 as
EB1 = 21198972 int119897
0int10
119887 [119867 () + 1198671015840 () (1 + 119868minus12 minus )]ℎ () + ℎ1015840 () (1 + 119868minus12 minus ) 119889119886 119889119887
= 21198972 int119897
0119887int10
(1198681 + 119887) + (119898 minus 1198731 + 119886) [1 + (1198681 + 119887) (1 minus )](1198681 + 119887) + (119898 minus 1198731 + 119886 minus 1) [1 + (1198681 + 119887) (1 minus )]119889119886 119889119887 = +1198972 (1198761 minus 1198762)
EB2 = 1119897 int119897
0int10
119867() + 1198671015840 () (1 + 119868minus12 minus )ℎ () + ℎ1015840 () (1 + 119868minus12 minus ) 119889119886 119889119887
= 119897 int119897
0int10
(1198681 + 119887) + (119898 minus 1198731 + 119886) [1 + (1198681 + 119887) (1 minus )](1198681 + 119887) + (119898 minus 1198731 + 119886 minus 1) [1 + (1198681 + 119887) (1 minus )]119889119886 119889119887 = +119897 (1198641 minus 1198642)
(25)
where
1198761 = [[1198972 minus (1198681 + 1198732119911 ())
2]] ln [(1198681 + 119897) 119911 () + 1198732]
+ (1198681 + 1198732119911 ())2
ln (1198681119911 () + 1198732) + 1198732119897119911 ()
1198762 = [[1198972 minus (1198681 + 1198732 minus 1119901 (1006704120582) )
2]]
sdot ln [(1198681 + 119897) 119901 (1006704120582) + 1198732 minus 1] + (1198681 + 1198732 minus 1119901 (1006704120582) )2
6 Mathematical Problems in Engineering
sdot ln (1198681119901 (1006704120582) + 1198732 minus 1) + (1198732 minus 1) 119897119901 (1006704120582) 1198641 = (1198681 + 119897 + 1198732119911 ()) ln [(1198681 + 119897) 119911 () + 1198732]minus (1198681 + 1198732119911 ()) ln [1198681119911 () + 1198732]
1198642 = (1198681 + 119897 + 1198732 minus 1119901 () ) ln [(1198681 + 119897) 119901 () + 1198732 minus 1]
minus (1198681 + 1198732 minus 1119901 () ) ln [1198681119901 () + 1198732 minus 1] 119911 () = + 1198732 (1 minus ) 119901 () = + (1198732 minus 1) (1 minus ) 1198732 = 119898 minus 1198731
(26)Similarly from (17) (21) and (24) the E-Bayesian estimate of120579119895 (119895 = 1 2) can be obtained as
119895EB1 =
2ℎ10158401198953 (1 minus ln120572119895119899 + 2120572119895119899 + 1) 119889lowast119895 = 119889119895
119889lowast119895 (1 minus ln120572119895119899 + 2120572119895119899 + 1) 119889lowast119895 = 119889119895
119895EB2 =
ℎ10158401198952 (1 minus ln120572119895119899 + 2120572119895119899 + 1) 119889lowast119895 = 119889119895
119889lowast119895 (1 minus ln120572119895119899 + 2120572119895119899 + 1) 119889lowast119895 = 119889119895
(27)
4 Application of the Pareto Distribution
In this section we analyze two real data sets to illustrate theapplication of the Pareto distribution in engineering practiceThe first data set (Set I) represent the failure time of 20mechanical components These data are taken from Murthyet al [25] and have been used by Bourguignon et al [22]These data are 0067 0068 0076 0081 0084 0085 00850086 0089 0098 0098 0114 0114 0115 0121 01250131 0149 0160 and 0485 The second data set (Set II) isgiven by Nelson [26] which represents the failure times tobreakdown of a type of electronic insulating material subjectto a constant-voltage stress These data are 035 059 096099 169 197 207 258 271 290 367 399 535 1377 and2550
We check the validity for the Pareto distribution 119875(120572 120579)by using the Kolmogorov test The statistics of the Kol-mogorov test are119863119899 = sup119909|119865(119909)minus119865119899(119909)| = max1le119894le20|119865(119909119894)minus119865119899(119909119894)| where 119865(119909) is the distribution function of 119875(120572 120579)and 119865(119909) = 1 minus (120579119909)120572 119909 gt 0 and 119865119899(119909) is the empiricaldistribution functions constructed based on the above dataSet I and Set II
For the data Set I when the parameters 120572 = 2113 and120579 = 0067 the observation value of the statistic is 119899 =max1le119894le20|119865(119909119894)minus119865119899(119909119894)| = 01303 For 119899 = 20 we choose thevalue of confidence level120572 as120572 = 001 002 005 respectivelyBy using the table of critical value on the Kolmogorov test weget the critical value 119899120572 where 20005 = 029403 20002 =032266 and 20001 = 035241 Since 119899 lt 20005 119899 lt20002 and 119899 lt 20001 thus the fit of Pareto distribution119875(120572 120579) = 119875(2113 0067) to the above data Set I is reasonable
For data Set II when the parameters 120572 = 051 and120579 = 035 the observation value of the statistic is 119899 =max1le119894le20|119865(119909119894)minus119865119899(119909119894)| = 02187 For 119899 = 15 we choose thevalue of confidence level120572 as120572 = 001 002 005 respectivelyBy using the table of critical value on the Kolmogorov testwe get the critical value 119899120572 where 15005 = 03376015002 = 037713 and 15001 = 040120 Since 119899 lt15005 119899 lt 15002 and 119899 lt 15001 the fit of Paretodistribution 119875(120572 120579) = 119875(051 035) to the above data Set IIis reasonable Therefore the use of the Pareto distribution iswell justified
5 Numerical Example
In this section we present numerical example using MonteCarlo simulations to illustrate the estimation procedurepresented in this paper The performance of three types ofBayesian estimates for different sample sizes and differentprogressive censoring schemes (PCS) is assessed based onsimulation dataThe term ldquodifferent PCSrdquomeans different setsof 119877119894rsquos In this paper we consider the following three differentPCS
Scheme 1 1198771 = sdot sdot sdot = 119877119898minus1 = 0 and 119877119898 = 119899 minus 119898Scheme 2 1198771 = 1198772 = sdot sdot sdot = 119877119871 = 0 119877119871+1 = sdot sdot sdot = 119877119898minus1 = 1and 119877119898 = 119899 minus 2119898 + 1 + 119871Scheme 3 1198771 = 1198772 = sdot sdot sdot = 119877119871+1 = 1 119877119871+2 = sdot sdot sdot = 119877119898minus1 =0 and 119877119898 = 119899 minus 119898 minus 1 minus 119871 where 119871 = [1198984] + 1 [1198984]representing the maximum integer that does not exceed1198984119898 ge 10
Before progressing further we first describe how togenerate Type II progressively censored competing failuredata from Pareto distribution under SSPALT
Step 1 Given119898 119899 and a certain censoring scheme generatetwo samples 1198801 1198802 119880119898 and 11988211198822 119882119898 from theuniform (0 1) distribution [27]
Step 2 Given 120572119895 120579119895 (119895 = 1 2) compute 11990511119894 and 11990512119894 bysolving equations 119880119894 = 1 minus 12057912057211 119905minus120572111119894 and 119882119894 = 1 minus 12057912057222 119905minus120572212119894 respectivelyThen record the minimum of (11990511119894 11990512119894) as 119905lowast119894 andthe corresponding index of the minimum (if 11990511119894 lt 11990512119894 set119888lowast119894 = 1 else set 119888lowast119894 = 2) for 1 le 119894 le 119898Step 3 For prefixed stress changing time 120591 find1198731 such that119905lowast1198731 le 120591 lt 119905lowast1198731+1 Then let 119905119894 = 119905lowast119894 and 119888119894 = 119888lowast119894 for 1 le 119894 le 1198731
Mathematical Problems in Engineering 7
Table 1 The AREs and MSEs of all estimates for differentm n and for censoring Scheme 1
(119899119898) Para OBEARE (MSE)
BEARE (MSE)
E-BE1ARE (MSE)
E-BE2ARE (MSE)
(40 10) 120582 00526 (00073) 00570 (00081) 00519 (00052) 00498 (0050)1205791 00360 (00048) 00378 (00076) 00392 (00075) 00382 (00065)1205792 00474 (00115) 00485 (00138) 00512 (00128) 00508 (00107)
(40 15) 120582 00514 (00069) 00551 (00067) 00515 (00048) 00495 (00044)1205791 00353 (00042) 00364 (00065) 00387 (00062) 00374 (00061)1205792 00462 (00107) 00472 (00120) 00504 (00112) 00500 (00099)
(40 20) 120582 00489 (00057) 00548 (00059) 00505 (00044) 00495 (00041)1205791 00336 (00032) 00341 (00056) 00372 (00051) 00365 (00049)1205792 00458 (00104) 00463 (00113) 00500 (00110) 00491 (00095)
(45 10) 120582 00513 (00070) 00564 (00080) 00514 (00050) 00498 (00048)1205791 00357 (00046) 00375 (00074) 00390 (00071) 00383 (00063)1205792 00471 (00111) 00486 (00127) 00505 (00115) 00512 (00100)
(45 15) 120582 00510 (00064) 00549 (00058) 00514 (00047) 00491 (00046)1205791 00350 (00040) 00362 (00061) 00386 (00059) 00370 (00060)1205792 00460 (00101) 00470 (00115) 00501 (00110) 00500 (00098)
(45 20) 120582 00484 (00056) 00542 (00054) 00504 (00041) 00493 (00038)1205791 00334 (00030) 00338 (00051) 00370 (00048) 00362 (00048)1205792 00454 (00100) 00462 (00111) 00497 (00105) 00489 (00087)
(50 10) 120582 00511 (00068) 00563 (00076) 00511 (00047) 00494 (00046)1205791 00353 (00044) 00373 (00071) 00387 (00070) 00380 (00061)1205792 00468 (00109) 00483 (00118) 00502 (00112) 00510 (00094)
(50 15) 120582 00509 (00062) 00546 (00058) 00512 (00045) 00492 (00045)1205791 00348 (00037) 00359 (00060) 00382 (00058) 00368 (00058)1205792 00459 (00095) 00468 (00110) 00500 (00107) 00500 (00093)
(50 20) 120582 00481 (00054) 00541 (00051) 00503 (00040) 00491 (00035)1205791 00330 (00027) 00335 (00047) 00368 (00042) 00369 (00043)1205792 00445 (00096) 00462 (00106) 00497 (00102) 00488 (00072)
Step 4 For 1198731 + 1 le 119894 le 119898 generate 11990521119894 by solvingthe equation 119880119894 = 1 minus 12057912057211 1205911205721(120582minus1)119905minus120572112058221119894 and generate 11990522119894 bysolving119882119894 = 1 minus 12057912057222 1205911205722(120582minus1)119905minus120572212058222119894 Again take the minimumof (11990521119894 11990522119894) as 119905lowast119894 as well as the corresponding index of theminimum (if 11990521119894 lt 11990522119894 set 119888lowast119894 = 1 else set 119888lowast119894 = 2)Step 5 For1198731 + 1 le 119894 le 119898 let 119905119894 = 119905lowast119894 and 119888119894 = 119888lowast119894
Then (1199051 1198881 1198771) (119905119898 119888119898 119877119898) is the required sample
We assumed that the occurrence time of the jth (119895 = 1 2)failure cause of tested product follows Pareto distributions119875(120572 120579) = 119875(2113 0067) and 119875(120572 120579) = 119875(182 0051) underuse stress levels 1198780 and 120582 = 20 and 120591 = 011 the parametersof prior density are taken as 119886 = 08 119887 = 20 1205731 = 0851205732 = 06 1198891 = 18 and 1198892 = 21 Considering different119898 119899 and censoring schemes we generate samples accordingto the above steps We replicate the process of evaluating1000 times in each case and compute the average relativeerrors (AREs) and mean squared errors (MSEs) ARE andMSE of the estimation of the parameter 120579 can be calculatedaccording to the following formulas
ARE () = 110001000sum119896=1
1003816100381610038161003816100381610038161003816120579 minus (119896)1003816100381610038161003816100381610038161003816120579
MSE () = 110001000sum119896=1
(120579 minus (119896))2 (28)
where (119896) is the 119896th estimator for the parameter 120579We calculate the AREs and MSEs of the objective
Bayesian estimates (OBE) Bayesian estimates (BE) andE-Bayesian estimates (E-BE) respectively The simulationresults are presented in Tables 1ndash3 In Tables 1ndash3 we denotethe E-BE based on the first prior as E-BE1 and the E-BE basedon the second prior as E-BE2
FromTables 1ndash3 the following observations can bemade
(1) For fixed n as m increases the AREs and MSEsdecrease for all estimates
(2) In terms of the values of AREs and MSEs the OBEperformances are better than that of BE and E-BE2
8 Mathematical Problems in Engineering
Table 2 The AREs and MSEs of all estimates for differentm n and for censoring Scheme 2
(119899 119898) Para OBEARE (MSE)
BEARE (MSE)
E-BE1ARE (MSE)
E-BE2ARE (MSE)
(40 10) 120582 00531 (00080) 00576 (00084) 00521 (00055) 00501 (00053)1205791 00365 (00051) 00381 (00081) 00398 (00079) 00386 (00067)1205792 00478 (00118) 00488 (00142) 00511 (00125) 00516 (00117)
(40 15) 120582 00516 (00071) 00551 (00082) 00515 (00051) 00495 (00049)1205791 00357 (00046) 00364 (00080) 00387 (00064) 00374 (00061)1205792 00465 (00113) 00472 (00138) 00504 (00114) 00500 (00112)
(40 20) 120582 00489 (00077) 00548 (00078) 00505 (00058) 00495 (00056)1205791 00336 (00044) 00341 (00075) 00372 (00056) 00365 (00053)1205792 00458 (00110) 00463 (00133) 00500 (00112) 00491 (00110)
(45 10) 120582 00527 (00079) 00564 (00079) 00511 (00052) 00501 (00051)1205791 00365 (00048) 00380 (00077) 00394 (00075) 00382 (00065)1205792 00474 (00115) 00485 (00133) 00509 (00122) 00512 (00114)
(45 15) 120582 00513 (00067) 00549 (00075) 00513 (00050) 00492 (00048)1205791 00348 (00037) 00364 (00074) 00382 (00062) 00370 (00058)1205792 00464 (00112) 00470 (00131) 00501 (00108) 00487 (00107)
(45 20) 120582 00486 (00065) 00546 (00071) 00501 (00057) 00492 (00045)1205791 00332 (00034) 00341 (00070) 00372 (00056) 00365 (00051)1205792 00451 (00110) 00457 (00127) 00492 (00104) 00490 (00104)
(50 10) 120582 00521 (00075) 00560 (00076) 00508 (00046) 00500 (00043)1205791 00362 (00046) 00380 (00075) 00392 (00074) 00381 (00063)1205792 00471 (00113) 00483 (00131) 00504 (00120) 00510 (00112)
(50 15) 120582 00504 (00065) 00546 (00074) 00512 (00050) 00491 (00045)1205791 00354 (00036) 00364 (00072) 00382 (00061) 00370 (00055)1205792 00464 (00110) 00470 (00130) 00501 (00105) 00496 (00104)
(50 20) 120582 00475 (00059) 00536 (00072) 00491 (00048) 00487 (00044)1205791 00324 (00029) 00332 (00170) 00366 (00052) 00356 (00051)1205792 00443 (00107) 00450 (00127) 00483 (00102) 00481 (00100)
performances are better than that of E-BE1 under thedifferent progressive censoring schemes
(3) Under the different progressive censoring schemesfor all estimates the values of AREs are less than 006and the values of MSEs are less than 0015 Hence theresults of all estimates are satisfied
6 Conclusions
In this paper the Bayesian analysis for the step-stress partiallyaccelerated competing failuremodel fromPareto distributionis investigated under Type II progressive censoring Based onthe noninformative prior and informative prior distributionswe derive the objective Bayesian estimates and Bayesianestimates of the unknown parameters and acceleration factorrespectively Considering such case that the hyperparametersare unknown and follow certain distributions the E-Bayesianmethod is utilized to estimate the unknown parameters andacceleration factor We analyze two real data sets to illustratethe application of the Pareto distribution in engineering
practice Finally the comparisons of the efficiencies of theobjective Bayesian estimates Bayesian estimates and corre-sponding E-Bayesian estimates are conducted It is concludedthat the OBE performances are better than that of BE andE-BE2 performances are better than that of E-BE1 underthe different progressive censoring schemes In addition wediscuss the effects of sample sizes and different progressivecensoring schemes on the estimates The simulation resultsdemonstrate that for fixed n the AREs and MSEs of allestimates decrease with119898 increasing
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The work is supported by the National Natural ScienceFoundation of China (71401134 71571144 and 71171164)The Natural Science Basic Research Program of ShaanxiProvince (2015JM1003) and The International Cooperation
Mathematical Problems in Engineering 9
Table 3 The AREs and MSEs of all estimates for differentm n and for censoring Scheme 3
(119899 119898) Para OBEARE (MSE)
BEARE (MSE)
E-BE1ARE (MSE)
E-BE2ARE (MSE)
(40 10) 120582 00535 (00079) 00578 (00082) 00521 (00056) 00510 (00054)1205791 00358 (00053) 00385 (00083) 00399 (00082) 00390 (00072)1205792 00481 (00129) 00491 (00145) 00512 (00130) 00520 (00122)
(40 15) 120582 00523 (00072) 00551 (00080) 00520 (00052) 00501 (00051)1205791 00364 (00048) 00371 (00077) 00396 (00076) 00380 (00071)1205792 00471 (00122) 00480 (00142) 00511 (00129) 00512 (00120)
(40 20) 120582 00491 (00066) 00552 (00078) 00513 (00051) 00504 (00050)1205791 00342 (00043) 00350 (00075) 00381 (00071) 00373 (00068)1205792 00465 (00120) 00469 (00140) 00507 (00124) 00499 (00118)
(45 10) 120582 00532 (00075) 00571 (00079) 00521 (00052) 00504 (00049)1205791 00363 (00051) 00381 (00081) 00398 (00081) 00390 (00067)1205792 00480 (00127) 00491 (00141) 00512 (00128) 00520 (00120)
(45 15) 120582 00518 (00068) 00551 (00075) 00520 (00050) 00501 (00046)1205791 00358 (00050) 00371 (00075) 00391 (00076) 00381 (00064)1205792 00467 (00118) 00478 (00137) 00510 (00123) 00510 (00117)
(45 20) 120582 00491 (00060) 00552 (00072) 00513 (00047) 00450 (00042)1205791 00340 (00048) 00345 (00069) 00381 (00065) 00371 (00062)1205792 00462 (00115) 00470 (00129) 00503 (00120) 00493 (00113)
(50 10) 120582 00530 (00074) 00571 (00067) 00520 (00050) 00502 (00045)1205791 00353 (00050) 00373 (00078) 00387 (00079) 00380 (00064)1205792 00468 (00124) 00483 (00140) 00502 (00124) 00510 (00119)
(50 15) 120582 00510 (00067) 00546 (00064) 00512 (00056) 00492 (00051)1205791 00348 (00042) 00359 (00075) 00382 (00076) 00368 (00062)1205792 00459 (00120) 00468 (00137) 00500 (00121) 00500 (00115)
(50 20) 120582 00481 (00065) 00541 (00062) 00503 (00053) 00491 (00050)1205791 00330 (00040) 00335 (00072) 00368 (00073) 00369 (00058)1205792 00454 (00116) 00462 (00134) 00497 (00118) 00485 (00113)
and Exchanges in Science and Technology Program ofShaanxi Province (2016KW-033)
References
[1] D Han and D Kundu ldquoInference for a step-stress model withcompeting risks for failure from the generalized exponentialdistribution under type-I censoringrdquo IEEE Transactions onReliability vol 64 no 1 pp 31ndash43 2015
[2] S Roy and C Mukhopadhyay ldquoMaximum likelihood analysisof multi-stress accelerated life test data of series systems withcompeting log-normal causes of failurerdquo Journal of Risk andReliability vol 229 no 2 pp 119ndash130 2015
[3] X P Zhang J Z Shang X Chen C H Zhang and Y S WangldquoStatistical inference of accelerated life testing with dependentcompeting failures based on copula theoryrdquo IEEE Transactionson Reliability vol 63 no 3 pp 764ndash780 2014
[4] Y-M Shi L Jin C Wei and H-B Yue ldquoConstant-stressaccelerated life test with competing risks under progressivetype-II hybrid censoringrdquo Advanced Materials Research vol712ndash715 pp 2080ndash2083 2013
[5] A Xu and Y Tang ldquoObjective Bayesian analysis of acceleratedcompeting failure models under Type-I censoringrdquo Computa-tional Statistics andData Analysis vol 55 no 10 pp 2830ndash28392011
[6] MWu Y Shi and Y Sun ldquoInference for accelerated competingfailure models fromWeibull distribution under Type-I progres-sive hybrid censoringrdquo Journal of Computational and AppliedMathematics vol 263 pp 423ndash431 2014
[7] N Balakrishnan andD Han ldquoExact inference for a simple step-stress model with competing risks for failure from exponentialdistribution under type-II censoringrdquo Journal of StatisticalPlanning and Inference vol 138 no 12 pp 4172ndash4186 2008
[8] D Han and N Balakrishnan ldquoInference for a simple step-stressmodel with competing risks for failure from the exponentialdistribution under time constraintrdquo Computational Statisticsand Data Analysis vol 54 no 9 pp 2066ndash2081 2010
[9] A M Abd-Elfattah A S Hassan and S G Nassr ldquoEstimationin step-stress partially accelerated life tests for the Burr typeXII distribution using type I censoringrdquo StatisticalMethodologyvol 5 no 6 pp 502ndash514 2008
[10] A A Ismail ldquoBayes estimation of Gompertz distributionparameters and acceleration factor under partially accelerated
10 Mathematical Problems in Engineering
life tests with type-I censoringrdquo Journal of Statistical Computa-tion and Simulation vol 80 no 11 pp 1253ndash1264 2010
[11] A A Ismail ldquoEstimating the parameters ofWeibull distributionand the acceleration factor from hybrid partially accelerated lifetestrdquo Applied Mathematical Modelling vol 36 no 7 pp 2920ndash2925 2012
[12] T A Abushal and A A Soliman ldquoEstimating the Paretoparameters under progressive censoring data for constant-partially accelerated life testsrdquo Journal of Statistical Computationand Simulation vol 85 no 5 pp 917ndash934 2015
[13] A A Ismail and A M Sarhan ldquoOptimal design of step-stresslife test with progressively type-II censored exponential datardquoInternational Mathematical Forum vol 4 no 37ndash40 pp 1963ndash1976 2009
[14] H Jeffreys The Theory of Probability Oxford University PressLondon UK 1998
[15] Q Guan Y Tang and A Xu ldquoObjective Bayesian analysis forbivariate Marshall-Olkin exponential distributionrdquo Computa-tional Statistics and Data Analysis vol 64 pp 299ndash313 2013
[16] M Han ldquoE-Bayesian estimation of failure probability and itsapplicationrdquoMathematical and ComputerModelling vol 45 no9-10 pp 1272ndash1279 2007
[17] M Han ldquoE-Bayesian estimation and hierarchical Bayesian esti-mation of failure probabilityrdquo Communications in StatisticsmdashTheory and Methods vol 40 no 18 pp 3303ndash3314 2011
[18] N L Johnson S Kotz and N Balakrishnan ContinuousUnivariate Distributions vol 1 Wiley-Interscience New YorkNY USA 2nd edition 1994
[19] AA Soliman ldquoEstimations for Paretomodel using general pro-gressive censored data and asymmetric lossrdquo Communicationsin StatisticsTheory andMethods vol 37 no 8ndash10 pp 1353ndash13702008
[20] H T Davis and M L Feldstein ldquoThe generalized Pareto law asa model for progressively censored survival datardquo Biometrikavol 66 no 2 pp 299ndash306 1979
[21] C M Harris ldquoThe Pareto distribution as a queue servicedisciplinerdquoOperations Research vol 16 no 2 pp 307ndash313 1968
[22] M BourguignonH Saulo andRN Fernandez ldquoA newPareto-type distribution with applications in reliability and incomedatardquo Physica A Statistical Mechanics and Its Applications vol457 pp 166ndash175 2016
[23] M Doostparast M G Akbari and N Balakrishnan ldquoBayesiananalysis for the two-parameter Pareto distribution based onrecord values and timesrdquo Journal of Statistical Computation andSimulation vol 81 no 11 pp 1393ndash1403 2011
[24] J O Berger Statistical Decision Theory and Bayesian AnalysisSpringer New York NY USA 2nd edition 1985
[25] D N P Murthy M Xie and R Jiang ldquoWeibull modelsrdquo inWiley Series in Probability and Statistics John Wiley and SonsHoboken NJ USA 2004
[26] W B Nelson ldquoStatistical methods for accelerated life test datathe inverse power law modelrdquo General Electric Co Tech Rep71-C011 Schenectady New York NY USA 1970
[27] N Balakrishnan and R A Sandhu ldquoA simple simulationalalgorithm for generating progressive Type-II censored samplesrdquoThe American Statistician vol 49 no 2 pp 229ndash230 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
derived the maximum likelihood estimates and Bayesianestimates of the acceleration factor and unknown parametersfor different lifetime distributions Abushal and Soliman [12]studied the MLE and Bayesian estimate of the parameters forthe Pareto distribution of the second type under progressivecensoring data for constant PALT Ismail and Sarhan [13]discussed optimal design of step-stress life test with progres-sively Type II censored exponential data But few authors havestudied competing failure models from this perspective andespecially from the Bayesian viewpoint
The Bayesian method has certain accurate advantageswhen sample size is small When little knowledge about theprior information is obtained we can derive the objectiveBayesian estimates based on noninformative priors such asthe Jeffreys prior and reference prior For more details seeJeffreys [14] Guan et al [15] derived the objective Bayesianestimates of unknown parameters for bivariate Marshall-Olkin exponential distribution respectively Under Type Icensoring Xu and Tang [5] considered the objective Bayesianinference for the accelerated competing failure modelsInstead with historical data or experience of some expertsavailable the subjective Bayesian approach on the basis ofinformative prior distributions can be implemented whileif the hyperparameters of the informative prior distributionsare unknownwe are able to use E-Bayesianmethod proposedby Han [16 17] By assuming the hyperparameters followcertain prior distributions the E-Bayesian estimate can beobtained as the expectation of the Bayesian estimate of theunknown parameter
In life testing and reliability studies the experimentermay not always obtain complete information on failure timesfor all experimental units Thus censoring is common tobe performed A censoring scheme (CS) which can balancebetween total time spent for the experiment number ofunits used in the experiment and the efficiency of statisticalinference based on the results of the experiment is desirable[12] Compared to the conventional Type I and Type IIcensoring scheme Type II progressive censoring schemeprovides higher flexibility to the experimenter by allowingfor product to be removed at nonterminal time points Thisallowance may be desirable when some degradation-relatedinformation from live test specimens needs to be collectedand analyzed or when scarce testing facilities need to bereleased for other experimentation In recent years some ofresearchers engaged in reliability research using progressivelycensored data such as Abushal and Soliman [12] and Ismailand Sarhan [13]
The Pareto distribution has found widespread use as amodel for various socioeconomic phenomena see [18 19]The Pareto distribution has also been used in reliability andlifetime modelling (see [20ndash22]) In this paper we will con-sider Bayesian analysis for the step-stress partially acceleratedcompeting failuremodel fromPareto distribution under TypeII progressive censoring The rest of this paper is organizedas follows In Section 2 under Type II progressive censoringscheme a step-stress partially accelerated competing failuremodel from Pareto distribution is described and some basicassumptions are given In Section 3 we obtain the objectiveBayesian estimates of the acceleration factor and unknown
parameters based on the noninformative prior distributionFurthermore the Bayesian and corresponding E-Bayesianestimates are provided In Section 4 we provide two real datasets to justify the use of the Pareto distribution in engineeringpractice The simulation results of all proposed methods fordifferent sample sizes and for different progressive censoringschemes are presented in Section 5 and some conclusions arepresented in Section 6
2 Assumptions and Model Description
Throughout the paper we make the following basic assump-tions
(1) The failure of a product occurs only due to one of the2 independent competing failure causes with lifetimes1198791 and 1198791 Then the lifetime of the product is 119879 =min(1198791 1198792)
(2) Two stress levels 1198780 and 1198781 are used in SSPALT (1198780 lt1198781) where 1198780 is use stress level and 1198781 is acceleratedstress level
(3) Let 1198791119895 be the occurrence time of the 119895th failure causeunder stress levels 1198780 and 1198792119895 be the occurrence timeof the 119895th failure cause under stress levels 1198781 119895 =1 2 The times 1198791119895 and 1198792119895 are independent Also1198791119895 follows a Pareto distribution with the followinghazard rate function (HRF) and reliability function
1199031119895 (119905) = 120572119895119905 119905 ge 1205791198951198781119895 (119905) = 120579120572119895119895 119905minus120572119895 119905 ge 120579119895 119895 = 1 2
(1)
where 120579119895 gt 0 is the scale parameter and 120572119895 gt 0 is theshape parameter 119895 = 1 2
(4) Tampered failure ratemodel holds that is theHRF 1199031119895is proportional to 1199032119895 Thus the HRF of the jth failurecause in SSPALT can be expressed as
119903lowast119895 (119905) =1199031119895 (119905) = 120572119895119905 120579119895 le 119905 le 1205911199032119895 (119905) = 120572119895120582119905 119905 gt 120591 120582 gt 1 (2)
where 120591 is prefixed number 120591 isin (0infin) and 120582 is accel-eration factor The corresponding reliability functionis
119878lowast119895 (119905) = 1198781119895 (119905) = 120579120572119895119895 119905minus120572119895 120579119895 le 119905 le 1205911198782119895 (119905) = 120579120572119895119895 120591120572119895(120582minus1)119905minus120572119895120582 119905 gt 120591 120582 gt 1 (3)
Under Type II progressive censored scheme SSPALT isdescribed as follows 119899 identical products are placed on thetest under the stress level 1198780 At the time of the first failure1198771 products are randomly removed from the remaining 119899 minus 1products and the failure cause index 1198881 is recorded Similarlyat the time of the second failure 1198772 products are randomly
Mathematical Problems in Engineering 3
removed from the remaining 119899 minus 2 minus 1198771 products and thefailure cause index 1198882 is recordedThe test continues until timepoint 120591 the stress levels are changed from 1198780 to 1198781 Under thestress level 1198781 the remaining 119899 minus 1198731 minus 1198771 minus 1198772 minus sdot sdot sdot minus 1198771198731products continue to be tested Suppose that the total numberof failures under the stress level 1198780 is 1198731 At the time of the(1198731 + 1)th failure 1198771198731+1 products are randomly removedfrom the remaining 119899 minus (1198731 + 1) minus 1198771 minus 1198772 minus sdot sdot sdot minus 1198771198731products and the failure cause index 1198881198731+1 is recorded Thetest continues until themth failure is observed and the failurecause index 119888119898 is recorded At this time all the remaining119877119898 = 119899minus119898minus1198771minus1198772minussdot sdot sdotminus119877119898minus1 products are removed and thetest terminates where119898 119877119894 (119894 = 1 2 119898 minus 1) are prefixednumber (1198731 lt 119898 lt 119899) and 119888119894 isin 1 2 119894 = 1 2 119898 Wedenote the time of the ith failure by 119905119894 and assume that eachfailure time alongwith the failure cause index is observed andthere is at least one failure caused by the jth (119895 = 1 2) failurecause under stress level 119878119896 (119896 = 0 1) Therefore observeddata can be denoted as (1199051 1198881) (1199052 1198882) (119905119898 119888119898) Then thelikelihood function of the failure samples caused by the 119895thfailure cause is
119871119895prop 1198731prod119894=1
1199031119895120575119895(119888119894) (119905119894) 1198781+1198771198941119895 (119905119894) 119898prod119894=1198731+1
1199032119895120575119895(119888119894) (119905119894) 1198781+1198771198942119895 (119905119894)= 120572119895119899119895120579120572119895119899119895 120582119899211989511988011198951198802119895119882minus1205721198951205821 119882120572119895(120582minus1)2
(4)
where
120575119895 (119888119894) = 1 119888119894 = 1198950 119888119894 = 119895
1198801119895 = 1198731prod119894=1
119905minus120575119895(119888119894)minus120572119895(1+119877119894)119894 1198802119895 = 119898prod
119894=1198731+1
119905minus120575119895(119888119894)119894 1198821 = 119898prod
119894=1198731+1
119905(1+119877119894)119894 1198822 = 120591119860(119898)119899119895 = 1198991119895 + 11989921198951198991119895 = 1198731sum
119894=1
120575119895 (119888119894) 1198992119895 = 119898sum
119894=1198731+1
120575119895 (119888119894) 119860 (119898) = 119898sum
119894=1198731+1
(1 + 119877119894)
(5)
Based on assumptions (1) and (4) we derive the full-likelihood function as
119871 (119905 | 120582 1205791 1205792) = 2prod119895=1
119871119895 prop ( 2prod119895=1
120572119899119895119895 11988011198951198802119895120579120572119895119899119895 )sdot 120582119898minus1198731119882minus(1205721+1205722)1205821 119882(1205721+1205722)(120582minus1)2
(6)
3 Bayesian Analysis
In this section the Bayesian method is employed to estimatethe scale parameters and acceleration factor when the shapeparameters are known Depending on whether the priorinformation is available or not we obtain the objectiveBayesian estimates Bayesian estimates and E-Bayesian esti-mates of the unknown parameters and acceleration factorunder the squared loss function respectively
31 Objective Bayesian Estimates When little knowledgeabout the prior information is obtained the objectiveBayesian approach is an alternative way We assume that thejoint prior distribution of the parameters (120582 1205791 1205792) is
120587119900 (120582 1205791 1205792) prop 112058212057911205792 (7)
From (6) and (7) the posterior density functions of theparameters 120582 1205791 and 1205792 can be given by 120587119900(120582 1205791 1205792 | 119905) prop119871(119905 | 120582 1205791 1205792)120587119900(120582 1205791 1205792) Then the marginal posteriordensity functions of the parameters 120582 1205791 and 1205792 can beobtained respectively
120587119900 (120582 | 119905) = 1198631120582119898minus1198731minus1119882minus(1205721+1205722)1205821 119882(1205721+1205722)1205822 120582 gt 1120587119900 (1205791 | 119905) = 11986321205791205721119899minus11 0 lt 1205791 lt 11990511120587119900 (1205792 | 119905) = 11986331205791205722119899minus12 0 lt 1205792 lt 11990521
(8)
where 1199051198961 is the first-failure time caused by failure cause119896 (119896 = 1 2) and1198631 = [intinfin1 120582119898minus1198731minus1119882minus(1205721+1205722)1205821 119882(1205721+1205722)1205822 119889120582]minus11198632 = [int119905110 1205791205721119899minus11 1198891205791]minus1 and1198633 = [int119905210 1205791205721119899minus12 1198891205792]minus1
Under the squared loss function the objective Bayesianestimates of the parameters 120582 1205791 and 1205792 can be obtained as
OBS = 1198631 intinfin1120582119898minus1198731119882minus(1205721+1205722)1205821 119882(1205721+1205722)1205822 119889120582
119895OBS = 1205721198951198991199051198951 (120572119895119899 + 1)minus1 119895 = 1 2(9)
It is obvious to see that (9) does not have explicit form Sowe obtain the Bayesian estimate of 120582 by numerical methodLet 119891(120582) = 120582119898minus1198731minus1119882minus(1205721+1205722)1205821 119882(1205721+1205722)1205822 119881 = 120582(1205721 +1205722) log(11988211198822) and then
intinfin1120582119898minus1198731minus1119882minus(1205721+1205722)1205821 119882(1205721+1205722)1205822 119889120582 = intinfin
0119891 (120582) 119889120582
minus int10119891 (120582) 119889120582 = [(1205721 + 1205722) log(11988211198822)]
minus(119898minus1198731)
4 Mathematical Problems in Engineering
sdot [intinfin0119881119898minus1198731minus1119890minus119881119889119881 minus int119881lowast
0119881119898minus1198731minus1119890minus119881119889119881]
= [(1205721 + 1205722) log(11988211198822)]minus(119898minus1198731)
sdot [Γ (119898 minus 1198731) minus int119881lowast
0119881119898minus1198731minus1119890minus119881119889119881]
(10)
where 119881lowast = (1205721 + 1205722) log(11988211198822) Similarly we get
intinfin1120582119891 (120582) 119889120582 = intinfin
0120582119891 (120582) 119889120582 minus int1
0120582119891 (120582) 119889120582
= [(1205721 + 1205722) log(11988211198822)]minus(119898minus1198731+1)
sdot [Γ (119898 minus 1198731 + 1) minus int119881lowast
0119881119898minus1198731119890minus119881119889119881]
(11)
Thus under the squared loss function the objective Bayesianestimates of the parameters 120582 can be obtained as
OBS = 1198631 intinfin1120582119898minus1198731119882minus(1205721+1205722)1205821 119882(1205721+1205722)1205822 119889120582 = intinfin1 120582119891 (120582) 119889120582intinfin
1119891 (120582) 119889120582
= [(1205721 + 1205722) log (11988211198822)] [Γ (119898 minus 1198731 + 1) minus int119881lowast
0119881119898minus1198731119890minus119881119889119881]
[Γ (119898 minus 1198731) minus int119881lowast0 119881119898minus1198731minus1119890minus119881119889119881] (12)
About the integralsint119881lowast0119881119898minus1198731119890minus119881119889119881 andint119881lowast
0119881119898minus1198731minus1119890minus119881119889119881
the compound trapezoidal integral formula can be used toobtain integral values The specific steps are as follows theinterval [0 119881lowast] is divided into 119899 equal subintervals and eachlength of subinterval is ℎ = 119881lowast119899 Then from the compoundtrapezoidal integral formula we get
int119881lowast0119892 (119881) 119889119881asymp ℎ2 [119892 (0) + 2
119899minus1sum119896=1
119892(119881lowast119896119899 ) + 119892 (119881lowast)] int119881lowast0119891 (119881) 119889119881asymp ℎ2 [119891 (0) + 2
119899minus1sum119896=1
119891(119881lowast119896119899 ) + 119891 (119881lowast)]
(13)
where 119892(119881) = 119881119898minus1198731119890minus119881 and 119891(119881) = 119881119898minus1198731minus1119890minus119881 Theabove two integration results are substituted into (12) andthe objective Bayesian estimates of the parameters 120582 can beobtained
32 Bayesian Estimates With historical data or expertsrsquo expe-rience available the informative prior could be a betteroption Following Doostparast et al [23] a convenient priorfamily of distributions for 120579119895 (119895 = 1 2) is provided bythe power function distributions with probability densityfunction
120587119904 (120579119895) = 120573119895120579120573119895minus1119895 119889minus120573119895119895 0 lt 120579119895 lt 119889119895 119889119895 120573119895 gt 0 (14)
We consider the prior density function of 120582 as follows120587119904 (120582) = 119860120582119886minus1 exp minus119887120582 119886 119887 gt 0 120582 gt 1 (15)
where 119860 is the regularization factor
Then the marginal posterior density functions of theparameters 120582 1205791 and 1205792 are respectively120587119904 (120582 | 119905) prop 120582119898minus1198731+119886minus1sdot exp minus [(1205721 + 1205722) (ln1198821 minus ln1198822) + 119887] 120582
119886 119887 gt 0 120582 gt 1120587119904 (120579119895 | 119905) = (120572119895119899 + 120573119895) 120579
120572119895119899+120573119895minus1
119895
119889lowast(120572119895119899+120573119895)119895
0 lt 120579119895 lt 119889lowast119895 119889lowast119895 = min (1199051198951 119889119895)
(16)
where 1199051198951 is the first-failure time caused by the 119895th failurecause (119895 = 1 2)
Under the squared loss function the Bayesian estimate of120579119895 (119895 = 1 2) can be easily obtained as
119895BS = int119889lowast119895
0120579119895120587119878 (120579119895 | 119905) 119889120579119895 = (120572119895119899 + 120573119895) 119889lowast119895(120572119895119899 + 120573119895 + 1) (17)
To obtain the Bayesian estimate of 120582 we expand 119867(120582) =120582119898minus1198731+119886 and ℎ(120582) = 120582119898minus1198731+119886minus1 in Taylor series aroundthe maximum likelihood estimate to get the followingapproximation
119867(120582) asymp 119867() + 1198671015840 () (120582 minus ) ℎ (120582) asymp ℎ () + ℎ1015840 () (120582 minus ) (18)
where1198671015840(120582) = (119898minus1198731 +119886)120582119898minus1198731+119886minus1 and ℎ1015840(120582) = (119898minus1198731 +119886 minus 1)120582119898minus1198731+119886minus2
Mathematical Problems in Engineering 5
Therefore under the squared loss function the Bayesianestimate of 120582 can be approximated as
BS = 119867() + 1198671015840 () (1 + 119868minus11 minus )ℎ () + ℎ1015840 () (1 + 119868minus11 minus ) (19)
where 1198681 = (1205721 + 1205722)(ln1198821 minus ln1198822) + 11988733 E-Bayesian Estimates By assuming that the hyperparam-eters follow certain distributions the Bayesian estimates canbe obtained when the hyperparameters in prior distributionsare unknown However complicated integration is often ahard work Han [16 17] gave the definition of E-Bayesianestimation to overcome this problem So we will derivethe E-Bayesian estimates of the unknown parameters andacceleration factor in this subsection
Suppose that the hyperparameters 119886 and 119887 and 120573119895 and119889119895 (119895 = 1 2) have the joint prior density functions 120587119890(119886 119887)and 120587119890(120573119895 119889119895) respectively Then the E-Bayesian estimates of120582 and 120579119895 (119895 = 1 2) are
EB = ∬119863BS120587119890 (119886 119887) 119889119886 119889119887 (20)
119895EB = ∬119863119895BS120587119890 (120573119895 119889119895) 119889120573119895119889 (119889119895) (21)
where BS and 119895BS are the Bayesian estimates of 120582 and 120579119895 (119895 =1 2) respectivelyAccording to Han [16] 119886 and 119887 and 120573119895 and 119889119895 (119895 = 1 2)
should be selected to guarantee that 120587119904(120582) and 120587119904(120579119895) aredecreasing functions of 120582 and 120579119895 respectivelyThe derivativesof 120587119904(120582) with respect to 120582 and 120587119904(120579119895) with respect to 120579119895 are
120597120587119904 (120582)120597120582 prop 120582119886minus2 exp minus119887120582 (119886 minus 1 minus 119887120582) 120597120587119904 (120579119895)120597120579119895 = 120573119895 (120573119895 minus 1) 120579120573119895minus2119895 119889minus120573119895119895
(22)
When 0 lt 119886 lt 1 119887 gt 0 120597120587119904(120582)120597120582 lt 0 and when 0 lt 120573119895 lt 1119889119895 gt 0 120597120587119904(120579119895)120597120579119895 lt 0Thus120587119904(120582) and120587119904(120579119895) are decreasingfunctions of 120582 and 120579119895 respectively
Given that 0 lt 119886 lt 1 and 0 lt 120573119895 lt 1 thelarger 119887 and 119889119895 are the thinner the tail of 120587119904(120582) and 120587119904(120579119895)will be Berger [24] showed that the thinner tailed priordistribution often reduces the robustness of the Bayesianestimate Consequently 119887 and 119889119895 (119895 = 1 2) should begiven upper bounds 119897 and ℎ1015840119895 respectively We consider thefollowing two joint prior distributions of 119886 and 119887 and 120573119895 and119889119895 (119895 = 1 2) respectively
1205871 (119886 119887) = 21198871198972 0 lt 119886 lt 1 0 lt 119887 lt 1198971205872 (119886 119887) = 1119897 0 lt 119886 lt 1 0 lt 119887 lt 119897 (23)
1205871119895 (120573119895 119889119895) = 2119889119895ℎ10158402119895 0 lt 120573119895 lt 1 0 lt 119889119895 lt ℎ10158401198951205872119895 (120573119895 119889119895) = 1ℎ1015840119895 0 lt 120573119895 lt 1 0 lt 119889119895 lt ℎ1015840119895
(24)
From (19) (20) and (23) we derive the E-Bayesian estimatesof 120582 as
EB1 = 21198972 int119897
0int10
119887 [119867 () + 1198671015840 () (1 + 119868minus12 minus )]ℎ () + ℎ1015840 () (1 + 119868minus12 minus ) 119889119886 119889119887
= 21198972 int119897
0119887int10
(1198681 + 119887) + (119898 minus 1198731 + 119886) [1 + (1198681 + 119887) (1 minus )](1198681 + 119887) + (119898 minus 1198731 + 119886 minus 1) [1 + (1198681 + 119887) (1 minus )]119889119886 119889119887 = +1198972 (1198761 minus 1198762)
EB2 = 1119897 int119897
0int10
119867() + 1198671015840 () (1 + 119868minus12 minus )ℎ () + ℎ1015840 () (1 + 119868minus12 minus ) 119889119886 119889119887
= 119897 int119897
0int10
(1198681 + 119887) + (119898 minus 1198731 + 119886) [1 + (1198681 + 119887) (1 minus )](1198681 + 119887) + (119898 minus 1198731 + 119886 minus 1) [1 + (1198681 + 119887) (1 minus )]119889119886 119889119887 = +119897 (1198641 minus 1198642)
(25)
where
1198761 = [[1198972 minus (1198681 + 1198732119911 ())
2]] ln [(1198681 + 119897) 119911 () + 1198732]
+ (1198681 + 1198732119911 ())2
ln (1198681119911 () + 1198732) + 1198732119897119911 ()
1198762 = [[1198972 minus (1198681 + 1198732 minus 1119901 (1006704120582) )
2]]
sdot ln [(1198681 + 119897) 119901 (1006704120582) + 1198732 minus 1] + (1198681 + 1198732 minus 1119901 (1006704120582) )2
6 Mathematical Problems in Engineering
sdot ln (1198681119901 (1006704120582) + 1198732 minus 1) + (1198732 minus 1) 119897119901 (1006704120582) 1198641 = (1198681 + 119897 + 1198732119911 ()) ln [(1198681 + 119897) 119911 () + 1198732]minus (1198681 + 1198732119911 ()) ln [1198681119911 () + 1198732]
1198642 = (1198681 + 119897 + 1198732 minus 1119901 () ) ln [(1198681 + 119897) 119901 () + 1198732 minus 1]
minus (1198681 + 1198732 minus 1119901 () ) ln [1198681119901 () + 1198732 minus 1] 119911 () = + 1198732 (1 minus ) 119901 () = + (1198732 minus 1) (1 minus ) 1198732 = 119898 minus 1198731
(26)Similarly from (17) (21) and (24) the E-Bayesian estimate of120579119895 (119895 = 1 2) can be obtained as
119895EB1 =
2ℎ10158401198953 (1 minus ln120572119895119899 + 2120572119895119899 + 1) 119889lowast119895 = 119889119895
119889lowast119895 (1 minus ln120572119895119899 + 2120572119895119899 + 1) 119889lowast119895 = 119889119895
119895EB2 =
ℎ10158401198952 (1 minus ln120572119895119899 + 2120572119895119899 + 1) 119889lowast119895 = 119889119895
119889lowast119895 (1 minus ln120572119895119899 + 2120572119895119899 + 1) 119889lowast119895 = 119889119895
(27)
4 Application of the Pareto Distribution
In this section we analyze two real data sets to illustrate theapplication of the Pareto distribution in engineering practiceThe first data set (Set I) represent the failure time of 20mechanical components These data are taken from Murthyet al [25] and have been used by Bourguignon et al [22]These data are 0067 0068 0076 0081 0084 0085 00850086 0089 0098 0098 0114 0114 0115 0121 01250131 0149 0160 and 0485 The second data set (Set II) isgiven by Nelson [26] which represents the failure times tobreakdown of a type of electronic insulating material subjectto a constant-voltage stress These data are 035 059 096099 169 197 207 258 271 290 367 399 535 1377 and2550
We check the validity for the Pareto distribution 119875(120572 120579)by using the Kolmogorov test The statistics of the Kol-mogorov test are119863119899 = sup119909|119865(119909)minus119865119899(119909)| = max1le119894le20|119865(119909119894)minus119865119899(119909119894)| where 119865(119909) is the distribution function of 119875(120572 120579)and 119865(119909) = 1 minus (120579119909)120572 119909 gt 0 and 119865119899(119909) is the empiricaldistribution functions constructed based on the above dataSet I and Set II
For the data Set I when the parameters 120572 = 2113 and120579 = 0067 the observation value of the statistic is 119899 =max1le119894le20|119865(119909119894)minus119865119899(119909119894)| = 01303 For 119899 = 20 we choose thevalue of confidence level120572 as120572 = 001 002 005 respectivelyBy using the table of critical value on the Kolmogorov test weget the critical value 119899120572 where 20005 = 029403 20002 =032266 and 20001 = 035241 Since 119899 lt 20005 119899 lt20002 and 119899 lt 20001 thus the fit of Pareto distribution119875(120572 120579) = 119875(2113 0067) to the above data Set I is reasonable
For data Set II when the parameters 120572 = 051 and120579 = 035 the observation value of the statistic is 119899 =max1le119894le20|119865(119909119894)minus119865119899(119909119894)| = 02187 For 119899 = 15 we choose thevalue of confidence level120572 as120572 = 001 002 005 respectivelyBy using the table of critical value on the Kolmogorov testwe get the critical value 119899120572 where 15005 = 03376015002 = 037713 and 15001 = 040120 Since 119899 lt15005 119899 lt 15002 and 119899 lt 15001 the fit of Paretodistribution 119875(120572 120579) = 119875(051 035) to the above data Set IIis reasonable Therefore the use of the Pareto distribution iswell justified
5 Numerical Example
In this section we present numerical example using MonteCarlo simulations to illustrate the estimation procedurepresented in this paper The performance of three types ofBayesian estimates for different sample sizes and differentprogressive censoring schemes (PCS) is assessed based onsimulation dataThe term ldquodifferent PCSrdquomeans different setsof 119877119894rsquos In this paper we consider the following three differentPCS
Scheme 1 1198771 = sdot sdot sdot = 119877119898minus1 = 0 and 119877119898 = 119899 minus 119898Scheme 2 1198771 = 1198772 = sdot sdot sdot = 119877119871 = 0 119877119871+1 = sdot sdot sdot = 119877119898minus1 = 1and 119877119898 = 119899 minus 2119898 + 1 + 119871Scheme 3 1198771 = 1198772 = sdot sdot sdot = 119877119871+1 = 1 119877119871+2 = sdot sdot sdot = 119877119898minus1 =0 and 119877119898 = 119899 minus 119898 minus 1 minus 119871 where 119871 = [1198984] + 1 [1198984]representing the maximum integer that does not exceed1198984119898 ge 10
Before progressing further we first describe how togenerate Type II progressively censored competing failuredata from Pareto distribution under SSPALT
Step 1 Given119898 119899 and a certain censoring scheme generatetwo samples 1198801 1198802 119880119898 and 11988211198822 119882119898 from theuniform (0 1) distribution [27]
Step 2 Given 120572119895 120579119895 (119895 = 1 2) compute 11990511119894 and 11990512119894 bysolving equations 119880119894 = 1 minus 12057912057211 119905minus120572111119894 and 119882119894 = 1 minus 12057912057222 119905minus120572212119894 respectivelyThen record the minimum of (11990511119894 11990512119894) as 119905lowast119894 andthe corresponding index of the minimum (if 11990511119894 lt 11990512119894 set119888lowast119894 = 1 else set 119888lowast119894 = 2) for 1 le 119894 le 119898Step 3 For prefixed stress changing time 120591 find1198731 such that119905lowast1198731 le 120591 lt 119905lowast1198731+1 Then let 119905119894 = 119905lowast119894 and 119888119894 = 119888lowast119894 for 1 le 119894 le 1198731
Mathematical Problems in Engineering 7
Table 1 The AREs and MSEs of all estimates for differentm n and for censoring Scheme 1
(119899119898) Para OBEARE (MSE)
BEARE (MSE)
E-BE1ARE (MSE)
E-BE2ARE (MSE)
(40 10) 120582 00526 (00073) 00570 (00081) 00519 (00052) 00498 (0050)1205791 00360 (00048) 00378 (00076) 00392 (00075) 00382 (00065)1205792 00474 (00115) 00485 (00138) 00512 (00128) 00508 (00107)
(40 15) 120582 00514 (00069) 00551 (00067) 00515 (00048) 00495 (00044)1205791 00353 (00042) 00364 (00065) 00387 (00062) 00374 (00061)1205792 00462 (00107) 00472 (00120) 00504 (00112) 00500 (00099)
(40 20) 120582 00489 (00057) 00548 (00059) 00505 (00044) 00495 (00041)1205791 00336 (00032) 00341 (00056) 00372 (00051) 00365 (00049)1205792 00458 (00104) 00463 (00113) 00500 (00110) 00491 (00095)
(45 10) 120582 00513 (00070) 00564 (00080) 00514 (00050) 00498 (00048)1205791 00357 (00046) 00375 (00074) 00390 (00071) 00383 (00063)1205792 00471 (00111) 00486 (00127) 00505 (00115) 00512 (00100)
(45 15) 120582 00510 (00064) 00549 (00058) 00514 (00047) 00491 (00046)1205791 00350 (00040) 00362 (00061) 00386 (00059) 00370 (00060)1205792 00460 (00101) 00470 (00115) 00501 (00110) 00500 (00098)
(45 20) 120582 00484 (00056) 00542 (00054) 00504 (00041) 00493 (00038)1205791 00334 (00030) 00338 (00051) 00370 (00048) 00362 (00048)1205792 00454 (00100) 00462 (00111) 00497 (00105) 00489 (00087)
(50 10) 120582 00511 (00068) 00563 (00076) 00511 (00047) 00494 (00046)1205791 00353 (00044) 00373 (00071) 00387 (00070) 00380 (00061)1205792 00468 (00109) 00483 (00118) 00502 (00112) 00510 (00094)
(50 15) 120582 00509 (00062) 00546 (00058) 00512 (00045) 00492 (00045)1205791 00348 (00037) 00359 (00060) 00382 (00058) 00368 (00058)1205792 00459 (00095) 00468 (00110) 00500 (00107) 00500 (00093)
(50 20) 120582 00481 (00054) 00541 (00051) 00503 (00040) 00491 (00035)1205791 00330 (00027) 00335 (00047) 00368 (00042) 00369 (00043)1205792 00445 (00096) 00462 (00106) 00497 (00102) 00488 (00072)
Step 4 For 1198731 + 1 le 119894 le 119898 generate 11990521119894 by solvingthe equation 119880119894 = 1 minus 12057912057211 1205911205721(120582minus1)119905minus120572112058221119894 and generate 11990522119894 bysolving119882119894 = 1 minus 12057912057222 1205911205722(120582minus1)119905minus120572212058222119894 Again take the minimumof (11990521119894 11990522119894) as 119905lowast119894 as well as the corresponding index of theminimum (if 11990521119894 lt 11990522119894 set 119888lowast119894 = 1 else set 119888lowast119894 = 2)Step 5 For1198731 + 1 le 119894 le 119898 let 119905119894 = 119905lowast119894 and 119888119894 = 119888lowast119894
Then (1199051 1198881 1198771) (119905119898 119888119898 119877119898) is the required sample
We assumed that the occurrence time of the jth (119895 = 1 2)failure cause of tested product follows Pareto distributions119875(120572 120579) = 119875(2113 0067) and 119875(120572 120579) = 119875(182 0051) underuse stress levels 1198780 and 120582 = 20 and 120591 = 011 the parametersof prior density are taken as 119886 = 08 119887 = 20 1205731 = 0851205732 = 06 1198891 = 18 and 1198892 = 21 Considering different119898 119899 and censoring schemes we generate samples accordingto the above steps We replicate the process of evaluating1000 times in each case and compute the average relativeerrors (AREs) and mean squared errors (MSEs) ARE andMSE of the estimation of the parameter 120579 can be calculatedaccording to the following formulas
ARE () = 110001000sum119896=1
1003816100381610038161003816100381610038161003816120579 minus (119896)1003816100381610038161003816100381610038161003816120579
MSE () = 110001000sum119896=1
(120579 minus (119896))2 (28)
where (119896) is the 119896th estimator for the parameter 120579We calculate the AREs and MSEs of the objective
Bayesian estimates (OBE) Bayesian estimates (BE) andE-Bayesian estimates (E-BE) respectively The simulationresults are presented in Tables 1ndash3 In Tables 1ndash3 we denotethe E-BE based on the first prior as E-BE1 and the E-BE basedon the second prior as E-BE2
FromTables 1ndash3 the following observations can bemade
(1) For fixed n as m increases the AREs and MSEsdecrease for all estimates
(2) In terms of the values of AREs and MSEs the OBEperformances are better than that of BE and E-BE2
8 Mathematical Problems in Engineering
Table 2 The AREs and MSEs of all estimates for differentm n and for censoring Scheme 2
(119899 119898) Para OBEARE (MSE)
BEARE (MSE)
E-BE1ARE (MSE)
E-BE2ARE (MSE)
(40 10) 120582 00531 (00080) 00576 (00084) 00521 (00055) 00501 (00053)1205791 00365 (00051) 00381 (00081) 00398 (00079) 00386 (00067)1205792 00478 (00118) 00488 (00142) 00511 (00125) 00516 (00117)
(40 15) 120582 00516 (00071) 00551 (00082) 00515 (00051) 00495 (00049)1205791 00357 (00046) 00364 (00080) 00387 (00064) 00374 (00061)1205792 00465 (00113) 00472 (00138) 00504 (00114) 00500 (00112)
(40 20) 120582 00489 (00077) 00548 (00078) 00505 (00058) 00495 (00056)1205791 00336 (00044) 00341 (00075) 00372 (00056) 00365 (00053)1205792 00458 (00110) 00463 (00133) 00500 (00112) 00491 (00110)
(45 10) 120582 00527 (00079) 00564 (00079) 00511 (00052) 00501 (00051)1205791 00365 (00048) 00380 (00077) 00394 (00075) 00382 (00065)1205792 00474 (00115) 00485 (00133) 00509 (00122) 00512 (00114)
(45 15) 120582 00513 (00067) 00549 (00075) 00513 (00050) 00492 (00048)1205791 00348 (00037) 00364 (00074) 00382 (00062) 00370 (00058)1205792 00464 (00112) 00470 (00131) 00501 (00108) 00487 (00107)
(45 20) 120582 00486 (00065) 00546 (00071) 00501 (00057) 00492 (00045)1205791 00332 (00034) 00341 (00070) 00372 (00056) 00365 (00051)1205792 00451 (00110) 00457 (00127) 00492 (00104) 00490 (00104)
(50 10) 120582 00521 (00075) 00560 (00076) 00508 (00046) 00500 (00043)1205791 00362 (00046) 00380 (00075) 00392 (00074) 00381 (00063)1205792 00471 (00113) 00483 (00131) 00504 (00120) 00510 (00112)
(50 15) 120582 00504 (00065) 00546 (00074) 00512 (00050) 00491 (00045)1205791 00354 (00036) 00364 (00072) 00382 (00061) 00370 (00055)1205792 00464 (00110) 00470 (00130) 00501 (00105) 00496 (00104)
(50 20) 120582 00475 (00059) 00536 (00072) 00491 (00048) 00487 (00044)1205791 00324 (00029) 00332 (00170) 00366 (00052) 00356 (00051)1205792 00443 (00107) 00450 (00127) 00483 (00102) 00481 (00100)
performances are better than that of E-BE1 under thedifferent progressive censoring schemes
(3) Under the different progressive censoring schemesfor all estimates the values of AREs are less than 006and the values of MSEs are less than 0015 Hence theresults of all estimates are satisfied
6 Conclusions
In this paper the Bayesian analysis for the step-stress partiallyaccelerated competing failuremodel fromPareto distributionis investigated under Type II progressive censoring Based onthe noninformative prior and informative prior distributionswe derive the objective Bayesian estimates and Bayesianestimates of the unknown parameters and acceleration factorrespectively Considering such case that the hyperparametersare unknown and follow certain distributions the E-Bayesianmethod is utilized to estimate the unknown parameters andacceleration factor We analyze two real data sets to illustratethe application of the Pareto distribution in engineering
practice Finally the comparisons of the efficiencies of theobjective Bayesian estimates Bayesian estimates and corre-sponding E-Bayesian estimates are conducted It is concludedthat the OBE performances are better than that of BE andE-BE2 performances are better than that of E-BE1 underthe different progressive censoring schemes In addition wediscuss the effects of sample sizes and different progressivecensoring schemes on the estimates The simulation resultsdemonstrate that for fixed n the AREs and MSEs of allestimates decrease with119898 increasing
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The work is supported by the National Natural ScienceFoundation of China (71401134 71571144 and 71171164)The Natural Science Basic Research Program of ShaanxiProvince (2015JM1003) and The International Cooperation
Mathematical Problems in Engineering 9
Table 3 The AREs and MSEs of all estimates for differentm n and for censoring Scheme 3
(119899 119898) Para OBEARE (MSE)
BEARE (MSE)
E-BE1ARE (MSE)
E-BE2ARE (MSE)
(40 10) 120582 00535 (00079) 00578 (00082) 00521 (00056) 00510 (00054)1205791 00358 (00053) 00385 (00083) 00399 (00082) 00390 (00072)1205792 00481 (00129) 00491 (00145) 00512 (00130) 00520 (00122)
(40 15) 120582 00523 (00072) 00551 (00080) 00520 (00052) 00501 (00051)1205791 00364 (00048) 00371 (00077) 00396 (00076) 00380 (00071)1205792 00471 (00122) 00480 (00142) 00511 (00129) 00512 (00120)
(40 20) 120582 00491 (00066) 00552 (00078) 00513 (00051) 00504 (00050)1205791 00342 (00043) 00350 (00075) 00381 (00071) 00373 (00068)1205792 00465 (00120) 00469 (00140) 00507 (00124) 00499 (00118)
(45 10) 120582 00532 (00075) 00571 (00079) 00521 (00052) 00504 (00049)1205791 00363 (00051) 00381 (00081) 00398 (00081) 00390 (00067)1205792 00480 (00127) 00491 (00141) 00512 (00128) 00520 (00120)
(45 15) 120582 00518 (00068) 00551 (00075) 00520 (00050) 00501 (00046)1205791 00358 (00050) 00371 (00075) 00391 (00076) 00381 (00064)1205792 00467 (00118) 00478 (00137) 00510 (00123) 00510 (00117)
(45 20) 120582 00491 (00060) 00552 (00072) 00513 (00047) 00450 (00042)1205791 00340 (00048) 00345 (00069) 00381 (00065) 00371 (00062)1205792 00462 (00115) 00470 (00129) 00503 (00120) 00493 (00113)
(50 10) 120582 00530 (00074) 00571 (00067) 00520 (00050) 00502 (00045)1205791 00353 (00050) 00373 (00078) 00387 (00079) 00380 (00064)1205792 00468 (00124) 00483 (00140) 00502 (00124) 00510 (00119)
(50 15) 120582 00510 (00067) 00546 (00064) 00512 (00056) 00492 (00051)1205791 00348 (00042) 00359 (00075) 00382 (00076) 00368 (00062)1205792 00459 (00120) 00468 (00137) 00500 (00121) 00500 (00115)
(50 20) 120582 00481 (00065) 00541 (00062) 00503 (00053) 00491 (00050)1205791 00330 (00040) 00335 (00072) 00368 (00073) 00369 (00058)1205792 00454 (00116) 00462 (00134) 00497 (00118) 00485 (00113)
and Exchanges in Science and Technology Program ofShaanxi Province (2016KW-033)
References
[1] D Han and D Kundu ldquoInference for a step-stress model withcompeting risks for failure from the generalized exponentialdistribution under type-I censoringrdquo IEEE Transactions onReliability vol 64 no 1 pp 31ndash43 2015
[2] S Roy and C Mukhopadhyay ldquoMaximum likelihood analysisof multi-stress accelerated life test data of series systems withcompeting log-normal causes of failurerdquo Journal of Risk andReliability vol 229 no 2 pp 119ndash130 2015
[3] X P Zhang J Z Shang X Chen C H Zhang and Y S WangldquoStatistical inference of accelerated life testing with dependentcompeting failures based on copula theoryrdquo IEEE Transactionson Reliability vol 63 no 3 pp 764ndash780 2014
[4] Y-M Shi L Jin C Wei and H-B Yue ldquoConstant-stressaccelerated life test with competing risks under progressivetype-II hybrid censoringrdquo Advanced Materials Research vol712ndash715 pp 2080ndash2083 2013
[5] A Xu and Y Tang ldquoObjective Bayesian analysis of acceleratedcompeting failure models under Type-I censoringrdquo Computa-tional Statistics andData Analysis vol 55 no 10 pp 2830ndash28392011
[6] MWu Y Shi and Y Sun ldquoInference for accelerated competingfailure models fromWeibull distribution under Type-I progres-sive hybrid censoringrdquo Journal of Computational and AppliedMathematics vol 263 pp 423ndash431 2014
[7] N Balakrishnan andD Han ldquoExact inference for a simple step-stress model with competing risks for failure from exponentialdistribution under type-II censoringrdquo Journal of StatisticalPlanning and Inference vol 138 no 12 pp 4172ndash4186 2008
[8] D Han and N Balakrishnan ldquoInference for a simple step-stressmodel with competing risks for failure from the exponentialdistribution under time constraintrdquo Computational Statisticsand Data Analysis vol 54 no 9 pp 2066ndash2081 2010
[9] A M Abd-Elfattah A S Hassan and S G Nassr ldquoEstimationin step-stress partially accelerated life tests for the Burr typeXII distribution using type I censoringrdquo StatisticalMethodologyvol 5 no 6 pp 502ndash514 2008
[10] A A Ismail ldquoBayes estimation of Gompertz distributionparameters and acceleration factor under partially accelerated
10 Mathematical Problems in Engineering
life tests with type-I censoringrdquo Journal of Statistical Computa-tion and Simulation vol 80 no 11 pp 1253ndash1264 2010
[11] A A Ismail ldquoEstimating the parameters ofWeibull distributionand the acceleration factor from hybrid partially accelerated lifetestrdquo Applied Mathematical Modelling vol 36 no 7 pp 2920ndash2925 2012
[12] T A Abushal and A A Soliman ldquoEstimating the Paretoparameters under progressive censoring data for constant-partially accelerated life testsrdquo Journal of Statistical Computationand Simulation vol 85 no 5 pp 917ndash934 2015
[13] A A Ismail and A M Sarhan ldquoOptimal design of step-stresslife test with progressively type-II censored exponential datardquoInternational Mathematical Forum vol 4 no 37ndash40 pp 1963ndash1976 2009
[14] H Jeffreys The Theory of Probability Oxford University PressLondon UK 1998
[15] Q Guan Y Tang and A Xu ldquoObjective Bayesian analysis forbivariate Marshall-Olkin exponential distributionrdquo Computa-tional Statistics and Data Analysis vol 64 pp 299ndash313 2013
[16] M Han ldquoE-Bayesian estimation of failure probability and itsapplicationrdquoMathematical and ComputerModelling vol 45 no9-10 pp 1272ndash1279 2007
[17] M Han ldquoE-Bayesian estimation and hierarchical Bayesian esti-mation of failure probabilityrdquo Communications in StatisticsmdashTheory and Methods vol 40 no 18 pp 3303ndash3314 2011
[18] N L Johnson S Kotz and N Balakrishnan ContinuousUnivariate Distributions vol 1 Wiley-Interscience New YorkNY USA 2nd edition 1994
[19] AA Soliman ldquoEstimations for Paretomodel using general pro-gressive censored data and asymmetric lossrdquo Communicationsin StatisticsTheory andMethods vol 37 no 8ndash10 pp 1353ndash13702008
[20] H T Davis and M L Feldstein ldquoThe generalized Pareto law asa model for progressively censored survival datardquo Biometrikavol 66 no 2 pp 299ndash306 1979
[21] C M Harris ldquoThe Pareto distribution as a queue servicedisciplinerdquoOperations Research vol 16 no 2 pp 307ndash313 1968
[22] M BourguignonH Saulo andRN Fernandez ldquoA newPareto-type distribution with applications in reliability and incomedatardquo Physica A Statistical Mechanics and Its Applications vol457 pp 166ndash175 2016
[23] M Doostparast M G Akbari and N Balakrishnan ldquoBayesiananalysis for the two-parameter Pareto distribution based onrecord values and timesrdquo Journal of Statistical Computation andSimulation vol 81 no 11 pp 1393ndash1403 2011
[24] J O Berger Statistical Decision Theory and Bayesian AnalysisSpringer New York NY USA 2nd edition 1985
[25] D N P Murthy M Xie and R Jiang ldquoWeibull modelsrdquo inWiley Series in Probability and Statistics John Wiley and SonsHoboken NJ USA 2004
[26] W B Nelson ldquoStatistical methods for accelerated life test datathe inverse power law modelrdquo General Electric Co Tech Rep71-C011 Schenectady New York NY USA 1970
[27] N Balakrishnan and R A Sandhu ldquoA simple simulationalalgorithm for generating progressive Type-II censored samplesrdquoThe American Statistician vol 49 no 2 pp 229ndash230 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
removed from the remaining 119899 minus 2 minus 1198771 products and thefailure cause index 1198882 is recordedThe test continues until timepoint 120591 the stress levels are changed from 1198780 to 1198781 Under thestress level 1198781 the remaining 119899 minus 1198731 minus 1198771 minus 1198772 minus sdot sdot sdot minus 1198771198731products continue to be tested Suppose that the total numberof failures under the stress level 1198780 is 1198731 At the time of the(1198731 + 1)th failure 1198771198731+1 products are randomly removedfrom the remaining 119899 minus (1198731 + 1) minus 1198771 minus 1198772 minus sdot sdot sdot minus 1198771198731products and the failure cause index 1198881198731+1 is recorded Thetest continues until themth failure is observed and the failurecause index 119888119898 is recorded At this time all the remaining119877119898 = 119899minus119898minus1198771minus1198772minussdot sdot sdotminus119877119898minus1 products are removed and thetest terminates where119898 119877119894 (119894 = 1 2 119898 minus 1) are prefixednumber (1198731 lt 119898 lt 119899) and 119888119894 isin 1 2 119894 = 1 2 119898 Wedenote the time of the ith failure by 119905119894 and assume that eachfailure time alongwith the failure cause index is observed andthere is at least one failure caused by the jth (119895 = 1 2) failurecause under stress level 119878119896 (119896 = 0 1) Therefore observeddata can be denoted as (1199051 1198881) (1199052 1198882) (119905119898 119888119898) Then thelikelihood function of the failure samples caused by the 119895thfailure cause is
119871119895prop 1198731prod119894=1
1199031119895120575119895(119888119894) (119905119894) 1198781+1198771198941119895 (119905119894) 119898prod119894=1198731+1
1199032119895120575119895(119888119894) (119905119894) 1198781+1198771198942119895 (119905119894)= 120572119895119899119895120579120572119895119899119895 120582119899211989511988011198951198802119895119882minus1205721198951205821 119882120572119895(120582minus1)2
(4)
where
120575119895 (119888119894) = 1 119888119894 = 1198950 119888119894 = 119895
1198801119895 = 1198731prod119894=1
119905minus120575119895(119888119894)minus120572119895(1+119877119894)119894 1198802119895 = 119898prod
119894=1198731+1
119905minus120575119895(119888119894)119894 1198821 = 119898prod
119894=1198731+1
119905(1+119877119894)119894 1198822 = 120591119860(119898)119899119895 = 1198991119895 + 11989921198951198991119895 = 1198731sum
119894=1
120575119895 (119888119894) 1198992119895 = 119898sum
119894=1198731+1
120575119895 (119888119894) 119860 (119898) = 119898sum
119894=1198731+1
(1 + 119877119894)
(5)
Based on assumptions (1) and (4) we derive the full-likelihood function as
119871 (119905 | 120582 1205791 1205792) = 2prod119895=1
119871119895 prop ( 2prod119895=1
120572119899119895119895 11988011198951198802119895120579120572119895119899119895 )sdot 120582119898minus1198731119882minus(1205721+1205722)1205821 119882(1205721+1205722)(120582minus1)2
(6)
3 Bayesian Analysis
In this section the Bayesian method is employed to estimatethe scale parameters and acceleration factor when the shapeparameters are known Depending on whether the priorinformation is available or not we obtain the objectiveBayesian estimates Bayesian estimates and E-Bayesian esti-mates of the unknown parameters and acceleration factorunder the squared loss function respectively
31 Objective Bayesian Estimates When little knowledgeabout the prior information is obtained the objectiveBayesian approach is an alternative way We assume that thejoint prior distribution of the parameters (120582 1205791 1205792) is
120587119900 (120582 1205791 1205792) prop 112058212057911205792 (7)
From (6) and (7) the posterior density functions of theparameters 120582 1205791 and 1205792 can be given by 120587119900(120582 1205791 1205792 | 119905) prop119871(119905 | 120582 1205791 1205792)120587119900(120582 1205791 1205792) Then the marginal posteriordensity functions of the parameters 120582 1205791 and 1205792 can beobtained respectively
120587119900 (120582 | 119905) = 1198631120582119898minus1198731minus1119882minus(1205721+1205722)1205821 119882(1205721+1205722)1205822 120582 gt 1120587119900 (1205791 | 119905) = 11986321205791205721119899minus11 0 lt 1205791 lt 11990511120587119900 (1205792 | 119905) = 11986331205791205722119899minus12 0 lt 1205792 lt 11990521
(8)
where 1199051198961 is the first-failure time caused by failure cause119896 (119896 = 1 2) and1198631 = [intinfin1 120582119898minus1198731minus1119882minus(1205721+1205722)1205821 119882(1205721+1205722)1205822 119889120582]minus11198632 = [int119905110 1205791205721119899minus11 1198891205791]minus1 and1198633 = [int119905210 1205791205721119899minus12 1198891205792]minus1
Under the squared loss function the objective Bayesianestimates of the parameters 120582 1205791 and 1205792 can be obtained as
OBS = 1198631 intinfin1120582119898minus1198731119882minus(1205721+1205722)1205821 119882(1205721+1205722)1205822 119889120582
119895OBS = 1205721198951198991199051198951 (120572119895119899 + 1)minus1 119895 = 1 2(9)
It is obvious to see that (9) does not have explicit form Sowe obtain the Bayesian estimate of 120582 by numerical methodLet 119891(120582) = 120582119898minus1198731minus1119882minus(1205721+1205722)1205821 119882(1205721+1205722)1205822 119881 = 120582(1205721 +1205722) log(11988211198822) and then
intinfin1120582119898minus1198731minus1119882minus(1205721+1205722)1205821 119882(1205721+1205722)1205822 119889120582 = intinfin
0119891 (120582) 119889120582
minus int10119891 (120582) 119889120582 = [(1205721 + 1205722) log(11988211198822)]
minus(119898minus1198731)
4 Mathematical Problems in Engineering
sdot [intinfin0119881119898minus1198731minus1119890minus119881119889119881 minus int119881lowast
0119881119898minus1198731minus1119890minus119881119889119881]
= [(1205721 + 1205722) log(11988211198822)]minus(119898minus1198731)
sdot [Γ (119898 minus 1198731) minus int119881lowast
0119881119898minus1198731minus1119890minus119881119889119881]
(10)
where 119881lowast = (1205721 + 1205722) log(11988211198822) Similarly we get
intinfin1120582119891 (120582) 119889120582 = intinfin
0120582119891 (120582) 119889120582 minus int1
0120582119891 (120582) 119889120582
= [(1205721 + 1205722) log(11988211198822)]minus(119898minus1198731+1)
sdot [Γ (119898 minus 1198731 + 1) minus int119881lowast
0119881119898minus1198731119890minus119881119889119881]
(11)
Thus under the squared loss function the objective Bayesianestimates of the parameters 120582 can be obtained as
OBS = 1198631 intinfin1120582119898minus1198731119882minus(1205721+1205722)1205821 119882(1205721+1205722)1205822 119889120582 = intinfin1 120582119891 (120582) 119889120582intinfin
1119891 (120582) 119889120582
= [(1205721 + 1205722) log (11988211198822)] [Γ (119898 minus 1198731 + 1) minus int119881lowast
0119881119898minus1198731119890minus119881119889119881]
[Γ (119898 minus 1198731) minus int119881lowast0 119881119898minus1198731minus1119890minus119881119889119881] (12)
About the integralsint119881lowast0119881119898minus1198731119890minus119881119889119881 andint119881lowast
0119881119898minus1198731minus1119890minus119881119889119881
the compound trapezoidal integral formula can be used toobtain integral values The specific steps are as follows theinterval [0 119881lowast] is divided into 119899 equal subintervals and eachlength of subinterval is ℎ = 119881lowast119899 Then from the compoundtrapezoidal integral formula we get
int119881lowast0119892 (119881) 119889119881asymp ℎ2 [119892 (0) + 2
119899minus1sum119896=1
119892(119881lowast119896119899 ) + 119892 (119881lowast)] int119881lowast0119891 (119881) 119889119881asymp ℎ2 [119891 (0) + 2
119899minus1sum119896=1
119891(119881lowast119896119899 ) + 119891 (119881lowast)]
(13)
where 119892(119881) = 119881119898minus1198731119890minus119881 and 119891(119881) = 119881119898minus1198731minus1119890minus119881 Theabove two integration results are substituted into (12) andthe objective Bayesian estimates of the parameters 120582 can beobtained
32 Bayesian Estimates With historical data or expertsrsquo expe-rience available the informative prior could be a betteroption Following Doostparast et al [23] a convenient priorfamily of distributions for 120579119895 (119895 = 1 2) is provided bythe power function distributions with probability densityfunction
120587119904 (120579119895) = 120573119895120579120573119895minus1119895 119889minus120573119895119895 0 lt 120579119895 lt 119889119895 119889119895 120573119895 gt 0 (14)
We consider the prior density function of 120582 as follows120587119904 (120582) = 119860120582119886minus1 exp minus119887120582 119886 119887 gt 0 120582 gt 1 (15)
where 119860 is the regularization factor
Then the marginal posterior density functions of theparameters 120582 1205791 and 1205792 are respectively120587119904 (120582 | 119905) prop 120582119898minus1198731+119886minus1sdot exp minus [(1205721 + 1205722) (ln1198821 minus ln1198822) + 119887] 120582
119886 119887 gt 0 120582 gt 1120587119904 (120579119895 | 119905) = (120572119895119899 + 120573119895) 120579
120572119895119899+120573119895minus1
119895
119889lowast(120572119895119899+120573119895)119895
0 lt 120579119895 lt 119889lowast119895 119889lowast119895 = min (1199051198951 119889119895)
(16)
where 1199051198951 is the first-failure time caused by the 119895th failurecause (119895 = 1 2)
Under the squared loss function the Bayesian estimate of120579119895 (119895 = 1 2) can be easily obtained as
119895BS = int119889lowast119895
0120579119895120587119878 (120579119895 | 119905) 119889120579119895 = (120572119895119899 + 120573119895) 119889lowast119895(120572119895119899 + 120573119895 + 1) (17)
To obtain the Bayesian estimate of 120582 we expand 119867(120582) =120582119898minus1198731+119886 and ℎ(120582) = 120582119898minus1198731+119886minus1 in Taylor series aroundthe maximum likelihood estimate to get the followingapproximation
119867(120582) asymp 119867() + 1198671015840 () (120582 minus ) ℎ (120582) asymp ℎ () + ℎ1015840 () (120582 minus ) (18)
where1198671015840(120582) = (119898minus1198731 +119886)120582119898minus1198731+119886minus1 and ℎ1015840(120582) = (119898minus1198731 +119886 minus 1)120582119898minus1198731+119886minus2
Mathematical Problems in Engineering 5
Therefore under the squared loss function the Bayesianestimate of 120582 can be approximated as
BS = 119867() + 1198671015840 () (1 + 119868minus11 minus )ℎ () + ℎ1015840 () (1 + 119868minus11 minus ) (19)
where 1198681 = (1205721 + 1205722)(ln1198821 minus ln1198822) + 11988733 E-Bayesian Estimates By assuming that the hyperparam-eters follow certain distributions the Bayesian estimates canbe obtained when the hyperparameters in prior distributionsare unknown However complicated integration is often ahard work Han [16 17] gave the definition of E-Bayesianestimation to overcome this problem So we will derivethe E-Bayesian estimates of the unknown parameters andacceleration factor in this subsection
Suppose that the hyperparameters 119886 and 119887 and 120573119895 and119889119895 (119895 = 1 2) have the joint prior density functions 120587119890(119886 119887)and 120587119890(120573119895 119889119895) respectively Then the E-Bayesian estimates of120582 and 120579119895 (119895 = 1 2) are
EB = ∬119863BS120587119890 (119886 119887) 119889119886 119889119887 (20)
119895EB = ∬119863119895BS120587119890 (120573119895 119889119895) 119889120573119895119889 (119889119895) (21)
where BS and 119895BS are the Bayesian estimates of 120582 and 120579119895 (119895 =1 2) respectivelyAccording to Han [16] 119886 and 119887 and 120573119895 and 119889119895 (119895 = 1 2)
should be selected to guarantee that 120587119904(120582) and 120587119904(120579119895) aredecreasing functions of 120582 and 120579119895 respectivelyThe derivativesof 120587119904(120582) with respect to 120582 and 120587119904(120579119895) with respect to 120579119895 are
120597120587119904 (120582)120597120582 prop 120582119886minus2 exp minus119887120582 (119886 minus 1 minus 119887120582) 120597120587119904 (120579119895)120597120579119895 = 120573119895 (120573119895 minus 1) 120579120573119895minus2119895 119889minus120573119895119895
(22)
When 0 lt 119886 lt 1 119887 gt 0 120597120587119904(120582)120597120582 lt 0 and when 0 lt 120573119895 lt 1119889119895 gt 0 120597120587119904(120579119895)120597120579119895 lt 0Thus120587119904(120582) and120587119904(120579119895) are decreasingfunctions of 120582 and 120579119895 respectively
Given that 0 lt 119886 lt 1 and 0 lt 120573119895 lt 1 thelarger 119887 and 119889119895 are the thinner the tail of 120587119904(120582) and 120587119904(120579119895)will be Berger [24] showed that the thinner tailed priordistribution often reduces the robustness of the Bayesianestimate Consequently 119887 and 119889119895 (119895 = 1 2) should begiven upper bounds 119897 and ℎ1015840119895 respectively We consider thefollowing two joint prior distributions of 119886 and 119887 and 120573119895 and119889119895 (119895 = 1 2) respectively
1205871 (119886 119887) = 21198871198972 0 lt 119886 lt 1 0 lt 119887 lt 1198971205872 (119886 119887) = 1119897 0 lt 119886 lt 1 0 lt 119887 lt 119897 (23)
1205871119895 (120573119895 119889119895) = 2119889119895ℎ10158402119895 0 lt 120573119895 lt 1 0 lt 119889119895 lt ℎ10158401198951205872119895 (120573119895 119889119895) = 1ℎ1015840119895 0 lt 120573119895 lt 1 0 lt 119889119895 lt ℎ1015840119895
(24)
From (19) (20) and (23) we derive the E-Bayesian estimatesof 120582 as
EB1 = 21198972 int119897
0int10
119887 [119867 () + 1198671015840 () (1 + 119868minus12 minus )]ℎ () + ℎ1015840 () (1 + 119868minus12 minus ) 119889119886 119889119887
= 21198972 int119897
0119887int10
(1198681 + 119887) + (119898 minus 1198731 + 119886) [1 + (1198681 + 119887) (1 minus )](1198681 + 119887) + (119898 minus 1198731 + 119886 minus 1) [1 + (1198681 + 119887) (1 minus )]119889119886 119889119887 = +1198972 (1198761 minus 1198762)
EB2 = 1119897 int119897
0int10
119867() + 1198671015840 () (1 + 119868minus12 minus )ℎ () + ℎ1015840 () (1 + 119868minus12 minus ) 119889119886 119889119887
= 119897 int119897
0int10
(1198681 + 119887) + (119898 minus 1198731 + 119886) [1 + (1198681 + 119887) (1 minus )](1198681 + 119887) + (119898 minus 1198731 + 119886 minus 1) [1 + (1198681 + 119887) (1 minus )]119889119886 119889119887 = +119897 (1198641 minus 1198642)
(25)
where
1198761 = [[1198972 minus (1198681 + 1198732119911 ())
2]] ln [(1198681 + 119897) 119911 () + 1198732]
+ (1198681 + 1198732119911 ())2
ln (1198681119911 () + 1198732) + 1198732119897119911 ()
1198762 = [[1198972 minus (1198681 + 1198732 minus 1119901 (1006704120582) )
2]]
sdot ln [(1198681 + 119897) 119901 (1006704120582) + 1198732 minus 1] + (1198681 + 1198732 minus 1119901 (1006704120582) )2
6 Mathematical Problems in Engineering
sdot ln (1198681119901 (1006704120582) + 1198732 minus 1) + (1198732 minus 1) 119897119901 (1006704120582) 1198641 = (1198681 + 119897 + 1198732119911 ()) ln [(1198681 + 119897) 119911 () + 1198732]minus (1198681 + 1198732119911 ()) ln [1198681119911 () + 1198732]
1198642 = (1198681 + 119897 + 1198732 minus 1119901 () ) ln [(1198681 + 119897) 119901 () + 1198732 minus 1]
minus (1198681 + 1198732 minus 1119901 () ) ln [1198681119901 () + 1198732 minus 1] 119911 () = + 1198732 (1 minus ) 119901 () = + (1198732 minus 1) (1 minus ) 1198732 = 119898 minus 1198731
(26)Similarly from (17) (21) and (24) the E-Bayesian estimate of120579119895 (119895 = 1 2) can be obtained as
119895EB1 =
2ℎ10158401198953 (1 minus ln120572119895119899 + 2120572119895119899 + 1) 119889lowast119895 = 119889119895
119889lowast119895 (1 minus ln120572119895119899 + 2120572119895119899 + 1) 119889lowast119895 = 119889119895
119895EB2 =
ℎ10158401198952 (1 minus ln120572119895119899 + 2120572119895119899 + 1) 119889lowast119895 = 119889119895
119889lowast119895 (1 minus ln120572119895119899 + 2120572119895119899 + 1) 119889lowast119895 = 119889119895
(27)
4 Application of the Pareto Distribution
In this section we analyze two real data sets to illustrate theapplication of the Pareto distribution in engineering practiceThe first data set (Set I) represent the failure time of 20mechanical components These data are taken from Murthyet al [25] and have been used by Bourguignon et al [22]These data are 0067 0068 0076 0081 0084 0085 00850086 0089 0098 0098 0114 0114 0115 0121 01250131 0149 0160 and 0485 The second data set (Set II) isgiven by Nelson [26] which represents the failure times tobreakdown of a type of electronic insulating material subjectto a constant-voltage stress These data are 035 059 096099 169 197 207 258 271 290 367 399 535 1377 and2550
We check the validity for the Pareto distribution 119875(120572 120579)by using the Kolmogorov test The statistics of the Kol-mogorov test are119863119899 = sup119909|119865(119909)minus119865119899(119909)| = max1le119894le20|119865(119909119894)minus119865119899(119909119894)| where 119865(119909) is the distribution function of 119875(120572 120579)and 119865(119909) = 1 minus (120579119909)120572 119909 gt 0 and 119865119899(119909) is the empiricaldistribution functions constructed based on the above dataSet I and Set II
For the data Set I when the parameters 120572 = 2113 and120579 = 0067 the observation value of the statistic is 119899 =max1le119894le20|119865(119909119894)minus119865119899(119909119894)| = 01303 For 119899 = 20 we choose thevalue of confidence level120572 as120572 = 001 002 005 respectivelyBy using the table of critical value on the Kolmogorov test weget the critical value 119899120572 where 20005 = 029403 20002 =032266 and 20001 = 035241 Since 119899 lt 20005 119899 lt20002 and 119899 lt 20001 thus the fit of Pareto distribution119875(120572 120579) = 119875(2113 0067) to the above data Set I is reasonable
For data Set II when the parameters 120572 = 051 and120579 = 035 the observation value of the statistic is 119899 =max1le119894le20|119865(119909119894)minus119865119899(119909119894)| = 02187 For 119899 = 15 we choose thevalue of confidence level120572 as120572 = 001 002 005 respectivelyBy using the table of critical value on the Kolmogorov testwe get the critical value 119899120572 where 15005 = 03376015002 = 037713 and 15001 = 040120 Since 119899 lt15005 119899 lt 15002 and 119899 lt 15001 the fit of Paretodistribution 119875(120572 120579) = 119875(051 035) to the above data Set IIis reasonable Therefore the use of the Pareto distribution iswell justified
5 Numerical Example
In this section we present numerical example using MonteCarlo simulations to illustrate the estimation procedurepresented in this paper The performance of three types ofBayesian estimates for different sample sizes and differentprogressive censoring schemes (PCS) is assessed based onsimulation dataThe term ldquodifferent PCSrdquomeans different setsof 119877119894rsquos In this paper we consider the following three differentPCS
Scheme 1 1198771 = sdot sdot sdot = 119877119898minus1 = 0 and 119877119898 = 119899 minus 119898Scheme 2 1198771 = 1198772 = sdot sdot sdot = 119877119871 = 0 119877119871+1 = sdot sdot sdot = 119877119898minus1 = 1and 119877119898 = 119899 minus 2119898 + 1 + 119871Scheme 3 1198771 = 1198772 = sdot sdot sdot = 119877119871+1 = 1 119877119871+2 = sdot sdot sdot = 119877119898minus1 =0 and 119877119898 = 119899 minus 119898 minus 1 minus 119871 where 119871 = [1198984] + 1 [1198984]representing the maximum integer that does not exceed1198984119898 ge 10
Before progressing further we first describe how togenerate Type II progressively censored competing failuredata from Pareto distribution under SSPALT
Step 1 Given119898 119899 and a certain censoring scheme generatetwo samples 1198801 1198802 119880119898 and 11988211198822 119882119898 from theuniform (0 1) distribution [27]
Step 2 Given 120572119895 120579119895 (119895 = 1 2) compute 11990511119894 and 11990512119894 bysolving equations 119880119894 = 1 minus 12057912057211 119905minus120572111119894 and 119882119894 = 1 minus 12057912057222 119905minus120572212119894 respectivelyThen record the minimum of (11990511119894 11990512119894) as 119905lowast119894 andthe corresponding index of the minimum (if 11990511119894 lt 11990512119894 set119888lowast119894 = 1 else set 119888lowast119894 = 2) for 1 le 119894 le 119898Step 3 For prefixed stress changing time 120591 find1198731 such that119905lowast1198731 le 120591 lt 119905lowast1198731+1 Then let 119905119894 = 119905lowast119894 and 119888119894 = 119888lowast119894 for 1 le 119894 le 1198731
Mathematical Problems in Engineering 7
Table 1 The AREs and MSEs of all estimates for differentm n and for censoring Scheme 1
(119899119898) Para OBEARE (MSE)
BEARE (MSE)
E-BE1ARE (MSE)
E-BE2ARE (MSE)
(40 10) 120582 00526 (00073) 00570 (00081) 00519 (00052) 00498 (0050)1205791 00360 (00048) 00378 (00076) 00392 (00075) 00382 (00065)1205792 00474 (00115) 00485 (00138) 00512 (00128) 00508 (00107)
(40 15) 120582 00514 (00069) 00551 (00067) 00515 (00048) 00495 (00044)1205791 00353 (00042) 00364 (00065) 00387 (00062) 00374 (00061)1205792 00462 (00107) 00472 (00120) 00504 (00112) 00500 (00099)
(40 20) 120582 00489 (00057) 00548 (00059) 00505 (00044) 00495 (00041)1205791 00336 (00032) 00341 (00056) 00372 (00051) 00365 (00049)1205792 00458 (00104) 00463 (00113) 00500 (00110) 00491 (00095)
(45 10) 120582 00513 (00070) 00564 (00080) 00514 (00050) 00498 (00048)1205791 00357 (00046) 00375 (00074) 00390 (00071) 00383 (00063)1205792 00471 (00111) 00486 (00127) 00505 (00115) 00512 (00100)
(45 15) 120582 00510 (00064) 00549 (00058) 00514 (00047) 00491 (00046)1205791 00350 (00040) 00362 (00061) 00386 (00059) 00370 (00060)1205792 00460 (00101) 00470 (00115) 00501 (00110) 00500 (00098)
(45 20) 120582 00484 (00056) 00542 (00054) 00504 (00041) 00493 (00038)1205791 00334 (00030) 00338 (00051) 00370 (00048) 00362 (00048)1205792 00454 (00100) 00462 (00111) 00497 (00105) 00489 (00087)
(50 10) 120582 00511 (00068) 00563 (00076) 00511 (00047) 00494 (00046)1205791 00353 (00044) 00373 (00071) 00387 (00070) 00380 (00061)1205792 00468 (00109) 00483 (00118) 00502 (00112) 00510 (00094)
(50 15) 120582 00509 (00062) 00546 (00058) 00512 (00045) 00492 (00045)1205791 00348 (00037) 00359 (00060) 00382 (00058) 00368 (00058)1205792 00459 (00095) 00468 (00110) 00500 (00107) 00500 (00093)
(50 20) 120582 00481 (00054) 00541 (00051) 00503 (00040) 00491 (00035)1205791 00330 (00027) 00335 (00047) 00368 (00042) 00369 (00043)1205792 00445 (00096) 00462 (00106) 00497 (00102) 00488 (00072)
Step 4 For 1198731 + 1 le 119894 le 119898 generate 11990521119894 by solvingthe equation 119880119894 = 1 minus 12057912057211 1205911205721(120582minus1)119905minus120572112058221119894 and generate 11990522119894 bysolving119882119894 = 1 minus 12057912057222 1205911205722(120582minus1)119905minus120572212058222119894 Again take the minimumof (11990521119894 11990522119894) as 119905lowast119894 as well as the corresponding index of theminimum (if 11990521119894 lt 11990522119894 set 119888lowast119894 = 1 else set 119888lowast119894 = 2)Step 5 For1198731 + 1 le 119894 le 119898 let 119905119894 = 119905lowast119894 and 119888119894 = 119888lowast119894
Then (1199051 1198881 1198771) (119905119898 119888119898 119877119898) is the required sample
We assumed that the occurrence time of the jth (119895 = 1 2)failure cause of tested product follows Pareto distributions119875(120572 120579) = 119875(2113 0067) and 119875(120572 120579) = 119875(182 0051) underuse stress levels 1198780 and 120582 = 20 and 120591 = 011 the parametersof prior density are taken as 119886 = 08 119887 = 20 1205731 = 0851205732 = 06 1198891 = 18 and 1198892 = 21 Considering different119898 119899 and censoring schemes we generate samples accordingto the above steps We replicate the process of evaluating1000 times in each case and compute the average relativeerrors (AREs) and mean squared errors (MSEs) ARE andMSE of the estimation of the parameter 120579 can be calculatedaccording to the following formulas
ARE () = 110001000sum119896=1
1003816100381610038161003816100381610038161003816120579 minus (119896)1003816100381610038161003816100381610038161003816120579
MSE () = 110001000sum119896=1
(120579 minus (119896))2 (28)
where (119896) is the 119896th estimator for the parameter 120579We calculate the AREs and MSEs of the objective
Bayesian estimates (OBE) Bayesian estimates (BE) andE-Bayesian estimates (E-BE) respectively The simulationresults are presented in Tables 1ndash3 In Tables 1ndash3 we denotethe E-BE based on the first prior as E-BE1 and the E-BE basedon the second prior as E-BE2
FromTables 1ndash3 the following observations can bemade
(1) For fixed n as m increases the AREs and MSEsdecrease for all estimates
(2) In terms of the values of AREs and MSEs the OBEperformances are better than that of BE and E-BE2
8 Mathematical Problems in Engineering
Table 2 The AREs and MSEs of all estimates for differentm n and for censoring Scheme 2
(119899 119898) Para OBEARE (MSE)
BEARE (MSE)
E-BE1ARE (MSE)
E-BE2ARE (MSE)
(40 10) 120582 00531 (00080) 00576 (00084) 00521 (00055) 00501 (00053)1205791 00365 (00051) 00381 (00081) 00398 (00079) 00386 (00067)1205792 00478 (00118) 00488 (00142) 00511 (00125) 00516 (00117)
(40 15) 120582 00516 (00071) 00551 (00082) 00515 (00051) 00495 (00049)1205791 00357 (00046) 00364 (00080) 00387 (00064) 00374 (00061)1205792 00465 (00113) 00472 (00138) 00504 (00114) 00500 (00112)
(40 20) 120582 00489 (00077) 00548 (00078) 00505 (00058) 00495 (00056)1205791 00336 (00044) 00341 (00075) 00372 (00056) 00365 (00053)1205792 00458 (00110) 00463 (00133) 00500 (00112) 00491 (00110)
(45 10) 120582 00527 (00079) 00564 (00079) 00511 (00052) 00501 (00051)1205791 00365 (00048) 00380 (00077) 00394 (00075) 00382 (00065)1205792 00474 (00115) 00485 (00133) 00509 (00122) 00512 (00114)
(45 15) 120582 00513 (00067) 00549 (00075) 00513 (00050) 00492 (00048)1205791 00348 (00037) 00364 (00074) 00382 (00062) 00370 (00058)1205792 00464 (00112) 00470 (00131) 00501 (00108) 00487 (00107)
(45 20) 120582 00486 (00065) 00546 (00071) 00501 (00057) 00492 (00045)1205791 00332 (00034) 00341 (00070) 00372 (00056) 00365 (00051)1205792 00451 (00110) 00457 (00127) 00492 (00104) 00490 (00104)
(50 10) 120582 00521 (00075) 00560 (00076) 00508 (00046) 00500 (00043)1205791 00362 (00046) 00380 (00075) 00392 (00074) 00381 (00063)1205792 00471 (00113) 00483 (00131) 00504 (00120) 00510 (00112)
(50 15) 120582 00504 (00065) 00546 (00074) 00512 (00050) 00491 (00045)1205791 00354 (00036) 00364 (00072) 00382 (00061) 00370 (00055)1205792 00464 (00110) 00470 (00130) 00501 (00105) 00496 (00104)
(50 20) 120582 00475 (00059) 00536 (00072) 00491 (00048) 00487 (00044)1205791 00324 (00029) 00332 (00170) 00366 (00052) 00356 (00051)1205792 00443 (00107) 00450 (00127) 00483 (00102) 00481 (00100)
performances are better than that of E-BE1 under thedifferent progressive censoring schemes
(3) Under the different progressive censoring schemesfor all estimates the values of AREs are less than 006and the values of MSEs are less than 0015 Hence theresults of all estimates are satisfied
6 Conclusions
In this paper the Bayesian analysis for the step-stress partiallyaccelerated competing failuremodel fromPareto distributionis investigated under Type II progressive censoring Based onthe noninformative prior and informative prior distributionswe derive the objective Bayesian estimates and Bayesianestimates of the unknown parameters and acceleration factorrespectively Considering such case that the hyperparametersare unknown and follow certain distributions the E-Bayesianmethod is utilized to estimate the unknown parameters andacceleration factor We analyze two real data sets to illustratethe application of the Pareto distribution in engineering
practice Finally the comparisons of the efficiencies of theobjective Bayesian estimates Bayesian estimates and corre-sponding E-Bayesian estimates are conducted It is concludedthat the OBE performances are better than that of BE andE-BE2 performances are better than that of E-BE1 underthe different progressive censoring schemes In addition wediscuss the effects of sample sizes and different progressivecensoring schemes on the estimates The simulation resultsdemonstrate that for fixed n the AREs and MSEs of allestimates decrease with119898 increasing
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The work is supported by the National Natural ScienceFoundation of China (71401134 71571144 and 71171164)The Natural Science Basic Research Program of ShaanxiProvince (2015JM1003) and The International Cooperation
Mathematical Problems in Engineering 9
Table 3 The AREs and MSEs of all estimates for differentm n and for censoring Scheme 3
(119899 119898) Para OBEARE (MSE)
BEARE (MSE)
E-BE1ARE (MSE)
E-BE2ARE (MSE)
(40 10) 120582 00535 (00079) 00578 (00082) 00521 (00056) 00510 (00054)1205791 00358 (00053) 00385 (00083) 00399 (00082) 00390 (00072)1205792 00481 (00129) 00491 (00145) 00512 (00130) 00520 (00122)
(40 15) 120582 00523 (00072) 00551 (00080) 00520 (00052) 00501 (00051)1205791 00364 (00048) 00371 (00077) 00396 (00076) 00380 (00071)1205792 00471 (00122) 00480 (00142) 00511 (00129) 00512 (00120)
(40 20) 120582 00491 (00066) 00552 (00078) 00513 (00051) 00504 (00050)1205791 00342 (00043) 00350 (00075) 00381 (00071) 00373 (00068)1205792 00465 (00120) 00469 (00140) 00507 (00124) 00499 (00118)
(45 10) 120582 00532 (00075) 00571 (00079) 00521 (00052) 00504 (00049)1205791 00363 (00051) 00381 (00081) 00398 (00081) 00390 (00067)1205792 00480 (00127) 00491 (00141) 00512 (00128) 00520 (00120)
(45 15) 120582 00518 (00068) 00551 (00075) 00520 (00050) 00501 (00046)1205791 00358 (00050) 00371 (00075) 00391 (00076) 00381 (00064)1205792 00467 (00118) 00478 (00137) 00510 (00123) 00510 (00117)
(45 20) 120582 00491 (00060) 00552 (00072) 00513 (00047) 00450 (00042)1205791 00340 (00048) 00345 (00069) 00381 (00065) 00371 (00062)1205792 00462 (00115) 00470 (00129) 00503 (00120) 00493 (00113)
(50 10) 120582 00530 (00074) 00571 (00067) 00520 (00050) 00502 (00045)1205791 00353 (00050) 00373 (00078) 00387 (00079) 00380 (00064)1205792 00468 (00124) 00483 (00140) 00502 (00124) 00510 (00119)
(50 15) 120582 00510 (00067) 00546 (00064) 00512 (00056) 00492 (00051)1205791 00348 (00042) 00359 (00075) 00382 (00076) 00368 (00062)1205792 00459 (00120) 00468 (00137) 00500 (00121) 00500 (00115)
(50 20) 120582 00481 (00065) 00541 (00062) 00503 (00053) 00491 (00050)1205791 00330 (00040) 00335 (00072) 00368 (00073) 00369 (00058)1205792 00454 (00116) 00462 (00134) 00497 (00118) 00485 (00113)
and Exchanges in Science and Technology Program ofShaanxi Province (2016KW-033)
References
[1] D Han and D Kundu ldquoInference for a step-stress model withcompeting risks for failure from the generalized exponentialdistribution under type-I censoringrdquo IEEE Transactions onReliability vol 64 no 1 pp 31ndash43 2015
[2] S Roy and C Mukhopadhyay ldquoMaximum likelihood analysisof multi-stress accelerated life test data of series systems withcompeting log-normal causes of failurerdquo Journal of Risk andReliability vol 229 no 2 pp 119ndash130 2015
[3] X P Zhang J Z Shang X Chen C H Zhang and Y S WangldquoStatistical inference of accelerated life testing with dependentcompeting failures based on copula theoryrdquo IEEE Transactionson Reliability vol 63 no 3 pp 764ndash780 2014
[4] Y-M Shi L Jin C Wei and H-B Yue ldquoConstant-stressaccelerated life test with competing risks under progressivetype-II hybrid censoringrdquo Advanced Materials Research vol712ndash715 pp 2080ndash2083 2013
[5] A Xu and Y Tang ldquoObjective Bayesian analysis of acceleratedcompeting failure models under Type-I censoringrdquo Computa-tional Statistics andData Analysis vol 55 no 10 pp 2830ndash28392011
[6] MWu Y Shi and Y Sun ldquoInference for accelerated competingfailure models fromWeibull distribution under Type-I progres-sive hybrid censoringrdquo Journal of Computational and AppliedMathematics vol 263 pp 423ndash431 2014
[7] N Balakrishnan andD Han ldquoExact inference for a simple step-stress model with competing risks for failure from exponentialdistribution under type-II censoringrdquo Journal of StatisticalPlanning and Inference vol 138 no 12 pp 4172ndash4186 2008
[8] D Han and N Balakrishnan ldquoInference for a simple step-stressmodel with competing risks for failure from the exponentialdistribution under time constraintrdquo Computational Statisticsand Data Analysis vol 54 no 9 pp 2066ndash2081 2010
[9] A M Abd-Elfattah A S Hassan and S G Nassr ldquoEstimationin step-stress partially accelerated life tests for the Burr typeXII distribution using type I censoringrdquo StatisticalMethodologyvol 5 no 6 pp 502ndash514 2008
[10] A A Ismail ldquoBayes estimation of Gompertz distributionparameters and acceleration factor under partially accelerated
10 Mathematical Problems in Engineering
life tests with type-I censoringrdquo Journal of Statistical Computa-tion and Simulation vol 80 no 11 pp 1253ndash1264 2010
[11] A A Ismail ldquoEstimating the parameters ofWeibull distributionand the acceleration factor from hybrid partially accelerated lifetestrdquo Applied Mathematical Modelling vol 36 no 7 pp 2920ndash2925 2012
[12] T A Abushal and A A Soliman ldquoEstimating the Paretoparameters under progressive censoring data for constant-partially accelerated life testsrdquo Journal of Statistical Computationand Simulation vol 85 no 5 pp 917ndash934 2015
[13] A A Ismail and A M Sarhan ldquoOptimal design of step-stresslife test with progressively type-II censored exponential datardquoInternational Mathematical Forum vol 4 no 37ndash40 pp 1963ndash1976 2009
[14] H Jeffreys The Theory of Probability Oxford University PressLondon UK 1998
[15] Q Guan Y Tang and A Xu ldquoObjective Bayesian analysis forbivariate Marshall-Olkin exponential distributionrdquo Computa-tional Statistics and Data Analysis vol 64 pp 299ndash313 2013
[16] M Han ldquoE-Bayesian estimation of failure probability and itsapplicationrdquoMathematical and ComputerModelling vol 45 no9-10 pp 1272ndash1279 2007
[17] M Han ldquoE-Bayesian estimation and hierarchical Bayesian esti-mation of failure probabilityrdquo Communications in StatisticsmdashTheory and Methods vol 40 no 18 pp 3303ndash3314 2011
[18] N L Johnson S Kotz and N Balakrishnan ContinuousUnivariate Distributions vol 1 Wiley-Interscience New YorkNY USA 2nd edition 1994
[19] AA Soliman ldquoEstimations for Paretomodel using general pro-gressive censored data and asymmetric lossrdquo Communicationsin StatisticsTheory andMethods vol 37 no 8ndash10 pp 1353ndash13702008
[20] H T Davis and M L Feldstein ldquoThe generalized Pareto law asa model for progressively censored survival datardquo Biometrikavol 66 no 2 pp 299ndash306 1979
[21] C M Harris ldquoThe Pareto distribution as a queue servicedisciplinerdquoOperations Research vol 16 no 2 pp 307ndash313 1968
[22] M BourguignonH Saulo andRN Fernandez ldquoA newPareto-type distribution with applications in reliability and incomedatardquo Physica A Statistical Mechanics and Its Applications vol457 pp 166ndash175 2016
[23] M Doostparast M G Akbari and N Balakrishnan ldquoBayesiananalysis for the two-parameter Pareto distribution based onrecord values and timesrdquo Journal of Statistical Computation andSimulation vol 81 no 11 pp 1393ndash1403 2011
[24] J O Berger Statistical Decision Theory and Bayesian AnalysisSpringer New York NY USA 2nd edition 1985
[25] D N P Murthy M Xie and R Jiang ldquoWeibull modelsrdquo inWiley Series in Probability and Statistics John Wiley and SonsHoboken NJ USA 2004
[26] W B Nelson ldquoStatistical methods for accelerated life test datathe inverse power law modelrdquo General Electric Co Tech Rep71-C011 Schenectady New York NY USA 1970
[27] N Balakrishnan and R A Sandhu ldquoA simple simulationalalgorithm for generating progressive Type-II censored samplesrdquoThe American Statistician vol 49 no 2 pp 229ndash230 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
sdot [intinfin0119881119898minus1198731minus1119890minus119881119889119881 minus int119881lowast
0119881119898minus1198731minus1119890minus119881119889119881]
= [(1205721 + 1205722) log(11988211198822)]minus(119898minus1198731)
sdot [Γ (119898 minus 1198731) minus int119881lowast
0119881119898minus1198731minus1119890minus119881119889119881]
(10)
where 119881lowast = (1205721 + 1205722) log(11988211198822) Similarly we get
intinfin1120582119891 (120582) 119889120582 = intinfin
0120582119891 (120582) 119889120582 minus int1
0120582119891 (120582) 119889120582
= [(1205721 + 1205722) log(11988211198822)]minus(119898minus1198731+1)
sdot [Γ (119898 minus 1198731 + 1) minus int119881lowast
0119881119898minus1198731119890minus119881119889119881]
(11)
Thus under the squared loss function the objective Bayesianestimates of the parameters 120582 can be obtained as
OBS = 1198631 intinfin1120582119898minus1198731119882minus(1205721+1205722)1205821 119882(1205721+1205722)1205822 119889120582 = intinfin1 120582119891 (120582) 119889120582intinfin
1119891 (120582) 119889120582
= [(1205721 + 1205722) log (11988211198822)] [Γ (119898 minus 1198731 + 1) minus int119881lowast
0119881119898minus1198731119890minus119881119889119881]
[Γ (119898 minus 1198731) minus int119881lowast0 119881119898minus1198731minus1119890minus119881119889119881] (12)
About the integralsint119881lowast0119881119898minus1198731119890minus119881119889119881 andint119881lowast
0119881119898minus1198731minus1119890minus119881119889119881
the compound trapezoidal integral formula can be used toobtain integral values The specific steps are as follows theinterval [0 119881lowast] is divided into 119899 equal subintervals and eachlength of subinterval is ℎ = 119881lowast119899 Then from the compoundtrapezoidal integral formula we get
int119881lowast0119892 (119881) 119889119881asymp ℎ2 [119892 (0) + 2
119899minus1sum119896=1
119892(119881lowast119896119899 ) + 119892 (119881lowast)] int119881lowast0119891 (119881) 119889119881asymp ℎ2 [119891 (0) + 2
119899minus1sum119896=1
119891(119881lowast119896119899 ) + 119891 (119881lowast)]
(13)
where 119892(119881) = 119881119898minus1198731119890minus119881 and 119891(119881) = 119881119898minus1198731minus1119890minus119881 Theabove two integration results are substituted into (12) andthe objective Bayesian estimates of the parameters 120582 can beobtained
32 Bayesian Estimates With historical data or expertsrsquo expe-rience available the informative prior could be a betteroption Following Doostparast et al [23] a convenient priorfamily of distributions for 120579119895 (119895 = 1 2) is provided bythe power function distributions with probability densityfunction
120587119904 (120579119895) = 120573119895120579120573119895minus1119895 119889minus120573119895119895 0 lt 120579119895 lt 119889119895 119889119895 120573119895 gt 0 (14)
We consider the prior density function of 120582 as follows120587119904 (120582) = 119860120582119886minus1 exp minus119887120582 119886 119887 gt 0 120582 gt 1 (15)
where 119860 is the regularization factor
Then the marginal posterior density functions of theparameters 120582 1205791 and 1205792 are respectively120587119904 (120582 | 119905) prop 120582119898minus1198731+119886minus1sdot exp minus [(1205721 + 1205722) (ln1198821 minus ln1198822) + 119887] 120582
119886 119887 gt 0 120582 gt 1120587119904 (120579119895 | 119905) = (120572119895119899 + 120573119895) 120579
120572119895119899+120573119895minus1
119895
119889lowast(120572119895119899+120573119895)119895
0 lt 120579119895 lt 119889lowast119895 119889lowast119895 = min (1199051198951 119889119895)
(16)
where 1199051198951 is the first-failure time caused by the 119895th failurecause (119895 = 1 2)
Under the squared loss function the Bayesian estimate of120579119895 (119895 = 1 2) can be easily obtained as
119895BS = int119889lowast119895
0120579119895120587119878 (120579119895 | 119905) 119889120579119895 = (120572119895119899 + 120573119895) 119889lowast119895(120572119895119899 + 120573119895 + 1) (17)
To obtain the Bayesian estimate of 120582 we expand 119867(120582) =120582119898minus1198731+119886 and ℎ(120582) = 120582119898minus1198731+119886minus1 in Taylor series aroundthe maximum likelihood estimate to get the followingapproximation
119867(120582) asymp 119867() + 1198671015840 () (120582 minus ) ℎ (120582) asymp ℎ () + ℎ1015840 () (120582 minus ) (18)
where1198671015840(120582) = (119898minus1198731 +119886)120582119898minus1198731+119886minus1 and ℎ1015840(120582) = (119898minus1198731 +119886 minus 1)120582119898minus1198731+119886minus2
Mathematical Problems in Engineering 5
Therefore under the squared loss function the Bayesianestimate of 120582 can be approximated as
BS = 119867() + 1198671015840 () (1 + 119868minus11 minus )ℎ () + ℎ1015840 () (1 + 119868minus11 minus ) (19)
where 1198681 = (1205721 + 1205722)(ln1198821 minus ln1198822) + 11988733 E-Bayesian Estimates By assuming that the hyperparam-eters follow certain distributions the Bayesian estimates canbe obtained when the hyperparameters in prior distributionsare unknown However complicated integration is often ahard work Han [16 17] gave the definition of E-Bayesianestimation to overcome this problem So we will derivethe E-Bayesian estimates of the unknown parameters andacceleration factor in this subsection
Suppose that the hyperparameters 119886 and 119887 and 120573119895 and119889119895 (119895 = 1 2) have the joint prior density functions 120587119890(119886 119887)and 120587119890(120573119895 119889119895) respectively Then the E-Bayesian estimates of120582 and 120579119895 (119895 = 1 2) are
EB = ∬119863BS120587119890 (119886 119887) 119889119886 119889119887 (20)
119895EB = ∬119863119895BS120587119890 (120573119895 119889119895) 119889120573119895119889 (119889119895) (21)
where BS and 119895BS are the Bayesian estimates of 120582 and 120579119895 (119895 =1 2) respectivelyAccording to Han [16] 119886 and 119887 and 120573119895 and 119889119895 (119895 = 1 2)
should be selected to guarantee that 120587119904(120582) and 120587119904(120579119895) aredecreasing functions of 120582 and 120579119895 respectivelyThe derivativesof 120587119904(120582) with respect to 120582 and 120587119904(120579119895) with respect to 120579119895 are
120597120587119904 (120582)120597120582 prop 120582119886minus2 exp minus119887120582 (119886 minus 1 minus 119887120582) 120597120587119904 (120579119895)120597120579119895 = 120573119895 (120573119895 minus 1) 120579120573119895minus2119895 119889minus120573119895119895
(22)
When 0 lt 119886 lt 1 119887 gt 0 120597120587119904(120582)120597120582 lt 0 and when 0 lt 120573119895 lt 1119889119895 gt 0 120597120587119904(120579119895)120597120579119895 lt 0Thus120587119904(120582) and120587119904(120579119895) are decreasingfunctions of 120582 and 120579119895 respectively
Given that 0 lt 119886 lt 1 and 0 lt 120573119895 lt 1 thelarger 119887 and 119889119895 are the thinner the tail of 120587119904(120582) and 120587119904(120579119895)will be Berger [24] showed that the thinner tailed priordistribution often reduces the robustness of the Bayesianestimate Consequently 119887 and 119889119895 (119895 = 1 2) should begiven upper bounds 119897 and ℎ1015840119895 respectively We consider thefollowing two joint prior distributions of 119886 and 119887 and 120573119895 and119889119895 (119895 = 1 2) respectively
1205871 (119886 119887) = 21198871198972 0 lt 119886 lt 1 0 lt 119887 lt 1198971205872 (119886 119887) = 1119897 0 lt 119886 lt 1 0 lt 119887 lt 119897 (23)
1205871119895 (120573119895 119889119895) = 2119889119895ℎ10158402119895 0 lt 120573119895 lt 1 0 lt 119889119895 lt ℎ10158401198951205872119895 (120573119895 119889119895) = 1ℎ1015840119895 0 lt 120573119895 lt 1 0 lt 119889119895 lt ℎ1015840119895
(24)
From (19) (20) and (23) we derive the E-Bayesian estimatesof 120582 as
EB1 = 21198972 int119897
0int10
119887 [119867 () + 1198671015840 () (1 + 119868minus12 minus )]ℎ () + ℎ1015840 () (1 + 119868minus12 minus ) 119889119886 119889119887
= 21198972 int119897
0119887int10
(1198681 + 119887) + (119898 minus 1198731 + 119886) [1 + (1198681 + 119887) (1 minus )](1198681 + 119887) + (119898 minus 1198731 + 119886 minus 1) [1 + (1198681 + 119887) (1 minus )]119889119886 119889119887 = +1198972 (1198761 minus 1198762)
EB2 = 1119897 int119897
0int10
119867() + 1198671015840 () (1 + 119868minus12 minus )ℎ () + ℎ1015840 () (1 + 119868minus12 minus ) 119889119886 119889119887
= 119897 int119897
0int10
(1198681 + 119887) + (119898 minus 1198731 + 119886) [1 + (1198681 + 119887) (1 minus )](1198681 + 119887) + (119898 minus 1198731 + 119886 minus 1) [1 + (1198681 + 119887) (1 minus )]119889119886 119889119887 = +119897 (1198641 minus 1198642)
(25)
where
1198761 = [[1198972 minus (1198681 + 1198732119911 ())
2]] ln [(1198681 + 119897) 119911 () + 1198732]
+ (1198681 + 1198732119911 ())2
ln (1198681119911 () + 1198732) + 1198732119897119911 ()
1198762 = [[1198972 minus (1198681 + 1198732 minus 1119901 (1006704120582) )
2]]
sdot ln [(1198681 + 119897) 119901 (1006704120582) + 1198732 minus 1] + (1198681 + 1198732 minus 1119901 (1006704120582) )2
6 Mathematical Problems in Engineering
sdot ln (1198681119901 (1006704120582) + 1198732 minus 1) + (1198732 minus 1) 119897119901 (1006704120582) 1198641 = (1198681 + 119897 + 1198732119911 ()) ln [(1198681 + 119897) 119911 () + 1198732]minus (1198681 + 1198732119911 ()) ln [1198681119911 () + 1198732]
1198642 = (1198681 + 119897 + 1198732 minus 1119901 () ) ln [(1198681 + 119897) 119901 () + 1198732 minus 1]
minus (1198681 + 1198732 minus 1119901 () ) ln [1198681119901 () + 1198732 minus 1] 119911 () = + 1198732 (1 minus ) 119901 () = + (1198732 minus 1) (1 minus ) 1198732 = 119898 minus 1198731
(26)Similarly from (17) (21) and (24) the E-Bayesian estimate of120579119895 (119895 = 1 2) can be obtained as
119895EB1 =
2ℎ10158401198953 (1 minus ln120572119895119899 + 2120572119895119899 + 1) 119889lowast119895 = 119889119895
119889lowast119895 (1 minus ln120572119895119899 + 2120572119895119899 + 1) 119889lowast119895 = 119889119895
119895EB2 =
ℎ10158401198952 (1 minus ln120572119895119899 + 2120572119895119899 + 1) 119889lowast119895 = 119889119895
119889lowast119895 (1 minus ln120572119895119899 + 2120572119895119899 + 1) 119889lowast119895 = 119889119895
(27)
4 Application of the Pareto Distribution
In this section we analyze two real data sets to illustrate theapplication of the Pareto distribution in engineering practiceThe first data set (Set I) represent the failure time of 20mechanical components These data are taken from Murthyet al [25] and have been used by Bourguignon et al [22]These data are 0067 0068 0076 0081 0084 0085 00850086 0089 0098 0098 0114 0114 0115 0121 01250131 0149 0160 and 0485 The second data set (Set II) isgiven by Nelson [26] which represents the failure times tobreakdown of a type of electronic insulating material subjectto a constant-voltage stress These data are 035 059 096099 169 197 207 258 271 290 367 399 535 1377 and2550
We check the validity for the Pareto distribution 119875(120572 120579)by using the Kolmogorov test The statistics of the Kol-mogorov test are119863119899 = sup119909|119865(119909)minus119865119899(119909)| = max1le119894le20|119865(119909119894)minus119865119899(119909119894)| where 119865(119909) is the distribution function of 119875(120572 120579)and 119865(119909) = 1 minus (120579119909)120572 119909 gt 0 and 119865119899(119909) is the empiricaldistribution functions constructed based on the above dataSet I and Set II
For the data Set I when the parameters 120572 = 2113 and120579 = 0067 the observation value of the statistic is 119899 =max1le119894le20|119865(119909119894)minus119865119899(119909119894)| = 01303 For 119899 = 20 we choose thevalue of confidence level120572 as120572 = 001 002 005 respectivelyBy using the table of critical value on the Kolmogorov test weget the critical value 119899120572 where 20005 = 029403 20002 =032266 and 20001 = 035241 Since 119899 lt 20005 119899 lt20002 and 119899 lt 20001 thus the fit of Pareto distribution119875(120572 120579) = 119875(2113 0067) to the above data Set I is reasonable
For data Set II when the parameters 120572 = 051 and120579 = 035 the observation value of the statistic is 119899 =max1le119894le20|119865(119909119894)minus119865119899(119909119894)| = 02187 For 119899 = 15 we choose thevalue of confidence level120572 as120572 = 001 002 005 respectivelyBy using the table of critical value on the Kolmogorov testwe get the critical value 119899120572 where 15005 = 03376015002 = 037713 and 15001 = 040120 Since 119899 lt15005 119899 lt 15002 and 119899 lt 15001 the fit of Paretodistribution 119875(120572 120579) = 119875(051 035) to the above data Set IIis reasonable Therefore the use of the Pareto distribution iswell justified
5 Numerical Example
In this section we present numerical example using MonteCarlo simulations to illustrate the estimation procedurepresented in this paper The performance of three types ofBayesian estimates for different sample sizes and differentprogressive censoring schemes (PCS) is assessed based onsimulation dataThe term ldquodifferent PCSrdquomeans different setsof 119877119894rsquos In this paper we consider the following three differentPCS
Scheme 1 1198771 = sdot sdot sdot = 119877119898minus1 = 0 and 119877119898 = 119899 minus 119898Scheme 2 1198771 = 1198772 = sdot sdot sdot = 119877119871 = 0 119877119871+1 = sdot sdot sdot = 119877119898minus1 = 1and 119877119898 = 119899 minus 2119898 + 1 + 119871Scheme 3 1198771 = 1198772 = sdot sdot sdot = 119877119871+1 = 1 119877119871+2 = sdot sdot sdot = 119877119898minus1 =0 and 119877119898 = 119899 minus 119898 minus 1 minus 119871 where 119871 = [1198984] + 1 [1198984]representing the maximum integer that does not exceed1198984119898 ge 10
Before progressing further we first describe how togenerate Type II progressively censored competing failuredata from Pareto distribution under SSPALT
Step 1 Given119898 119899 and a certain censoring scheme generatetwo samples 1198801 1198802 119880119898 and 11988211198822 119882119898 from theuniform (0 1) distribution [27]
Step 2 Given 120572119895 120579119895 (119895 = 1 2) compute 11990511119894 and 11990512119894 bysolving equations 119880119894 = 1 minus 12057912057211 119905minus120572111119894 and 119882119894 = 1 minus 12057912057222 119905minus120572212119894 respectivelyThen record the minimum of (11990511119894 11990512119894) as 119905lowast119894 andthe corresponding index of the minimum (if 11990511119894 lt 11990512119894 set119888lowast119894 = 1 else set 119888lowast119894 = 2) for 1 le 119894 le 119898Step 3 For prefixed stress changing time 120591 find1198731 such that119905lowast1198731 le 120591 lt 119905lowast1198731+1 Then let 119905119894 = 119905lowast119894 and 119888119894 = 119888lowast119894 for 1 le 119894 le 1198731
Mathematical Problems in Engineering 7
Table 1 The AREs and MSEs of all estimates for differentm n and for censoring Scheme 1
(119899119898) Para OBEARE (MSE)
BEARE (MSE)
E-BE1ARE (MSE)
E-BE2ARE (MSE)
(40 10) 120582 00526 (00073) 00570 (00081) 00519 (00052) 00498 (0050)1205791 00360 (00048) 00378 (00076) 00392 (00075) 00382 (00065)1205792 00474 (00115) 00485 (00138) 00512 (00128) 00508 (00107)
(40 15) 120582 00514 (00069) 00551 (00067) 00515 (00048) 00495 (00044)1205791 00353 (00042) 00364 (00065) 00387 (00062) 00374 (00061)1205792 00462 (00107) 00472 (00120) 00504 (00112) 00500 (00099)
(40 20) 120582 00489 (00057) 00548 (00059) 00505 (00044) 00495 (00041)1205791 00336 (00032) 00341 (00056) 00372 (00051) 00365 (00049)1205792 00458 (00104) 00463 (00113) 00500 (00110) 00491 (00095)
(45 10) 120582 00513 (00070) 00564 (00080) 00514 (00050) 00498 (00048)1205791 00357 (00046) 00375 (00074) 00390 (00071) 00383 (00063)1205792 00471 (00111) 00486 (00127) 00505 (00115) 00512 (00100)
(45 15) 120582 00510 (00064) 00549 (00058) 00514 (00047) 00491 (00046)1205791 00350 (00040) 00362 (00061) 00386 (00059) 00370 (00060)1205792 00460 (00101) 00470 (00115) 00501 (00110) 00500 (00098)
(45 20) 120582 00484 (00056) 00542 (00054) 00504 (00041) 00493 (00038)1205791 00334 (00030) 00338 (00051) 00370 (00048) 00362 (00048)1205792 00454 (00100) 00462 (00111) 00497 (00105) 00489 (00087)
(50 10) 120582 00511 (00068) 00563 (00076) 00511 (00047) 00494 (00046)1205791 00353 (00044) 00373 (00071) 00387 (00070) 00380 (00061)1205792 00468 (00109) 00483 (00118) 00502 (00112) 00510 (00094)
(50 15) 120582 00509 (00062) 00546 (00058) 00512 (00045) 00492 (00045)1205791 00348 (00037) 00359 (00060) 00382 (00058) 00368 (00058)1205792 00459 (00095) 00468 (00110) 00500 (00107) 00500 (00093)
(50 20) 120582 00481 (00054) 00541 (00051) 00503 (00040) 00491 (00035)1205791 00330 (00027) 00335 (00047) 00368 (00042) 00369 (00043)1205792 00445 (00096) 00462 (00106) 00497 (00102) 00488 (00072)
Step 4 For 1198731 + 1 le 119894 le 119898 generate 11990521119894 by solvingthe equation 119880119894 = 1 minus 12057912057211 1205911205721(120582minus1)119905minus120572112058221119894 and generate 11990522119894 bysolving119882119894 = 1 minus 12057912057222 1205911205722(120582minus1)119905minus120572212058222119894 Again take the minimumof (11990521119894 11990522119894) as 119905lowast119894 as well as the corresponding index of theminimum (if 11990521119894 lt 11990522119894 set 119888lowast119894 = 1 else set 119888lowast119894 = 2)Step 5 For1198731 + 1 le 119894 le 119898 let 119905119894 = 119905lowast119894 and 119888119894 = 119888lowast119894
Then (1199051 1198881 1198771) (119905119898 119888119898 119877119898) is the required sample
We assumed that the occurrence time of the jth (119895 = 1 2)failure cause of tested product follows Pareto distributions119875(120572 120579) = 119875(2113 0067) and 119875(120572 120579) = 119875(182 0051) underuse stress levels 1198780 and 120582 = 20 and 120591 = 011 the parametersof prior density are taken as 119886 = 08 119887 = 20 1205731 = 0851205732 = 06 1198891 = 18 and 1198892 = 21 Considering different119898 119899 and censoring schemes we generate samples accordingto the above steps We replicate the process of evaluating1000 times in each case and compute the average relativeerrors (AREs) and mean squared errors (MSEs) ARE andMSE of the estimation of the parameter 120579 can be calculatedaccording to the following formulas
ARE () = 110001000sum119896=1
1003816100381610038161003816100381610038161003816120579 minus (119896)1003816100381610038161003816100381610038161003816120579
MSE () = 110001000sum119896=1
(120579 minus (119896))2 (28)
where (119896) is the 119896th estimator for the parameter 120579We calculate the AREs and MSEs of the objective
Bayesian estimates (OBE) Bayesian estimates (BE) andE-Bayesian estimates (E-BE) respectively The simulationresults are presented in Tables 1ndash3 In Tables 1ndash3 we denotethe E-BE based on the first prior as E-BE1 and the E-BE basedon the second prior as E-BE2
FromTables 1ndash3 the following observations can bemade
(1) For fixed n as m increases the AREs and MSEsdecrease for all estimates
(2) In terms of the values of AREs and MSEs the OBEperformances are better than that of BE and E-BE2
8 Mathematical Problems in Engineering
Table 2 The AREs and MSEs of all estimates for differentm n and for censoring Scheme 2
(119899 119898) Para OBEARE (MSE)
BEARE (MSE)
E-BE1ARE (MSE)
E-BE2ARE (MSE)
(40 10) 120582 00531 (00080) 00576 (00084) 00521 (00055) 00501 (00053)1205791 00365 (00051) 00381 (00081) 00398 (00079) 00386 (00067)1205792 00478 (00118) 00488 (00142) 00511 (00125) 00516 (00117)
(40 15) 120582 00516 (00071) 00551 (00082) 00515 (00051) 00495 (00049)1205791 00357 (00046) 00364 (00080) 00387 (00064) 00374 (00061)1205792 00465 (00113) 00472 (00138) 00504 (00114) 00500 (00112)
(40 20) 120582 00489 (00077) 00548 (00078) 00505 (00058) 00495 (00056)1205791 00336 (00044) 00341 (00075) 00372 (00056) 00365 (00053)1205792 00458 (00110) 00463 (00133) 00500 (00112) 00491 (00110)
(45 10) 120582 00527 (00079) 00564 (00079) 00511 (00052) 00501 (00051)1205791 00365 (00048) 00380 (00077) 00394 (00075) 00382 (00065)1205792 00474 (00115) 00485 (00133) 00509 (00122) 00512 (00114)
(45 15) 120582 00513 (00067) 00549 (00075) 00513 (00050) 00492 (00048)1205791 00348 (00037) 00364 (00074) 00382 (00062) 00370 (00058)1205792 00464 (00112) 00470 (00131) 00501 (00108) 00487 (00107)
(45 20) 120582 00486 (00065) 00546 (00071) 00501 (00057) 00492 (00045)1205791 00332 (00034) 00341 (00070) 00372 (00056) 00365 (00051)1205792 00451 (00110) 00457 (00127) 00492 (00104) 00490 (00104)
(50 10) 120582 00521 (00075) 00560 (00076) 00508 (00046) 00500 (00043)1205791 00362 (00046) 00380 (00075) 00392 (00074) 00381 (00063)1205792 00471 (00113) 00483 (00131) 00504 (00120) 00510 (00112)
(50 15) 120582 00504 (00065) 00546 (00074) 00512 (00050) 00491 (00045)1205791 00354 (00036) 00364 (00072) 00382 (00061) 00370 (00055)1205792 00464 (00110) 00470 (00130) 00501 (00105) 00496 (00104)
(50 20) 120582 00475 (00059) 00536 (00072) 00491 (00048) 00487 (00044)1205791 00324 (00029) 00332 (00170) 00366 (00052) 00356 (00051)1205792 00443 (00107) 00450 (00127) 00483 (00102) 00481 (00100)
performances are better than that of E-BE1 under thedifferent progressive censoring schemes
(3) Under the different progressive censoring schemesfor all estimates the values of AREs are less than 006and the values of MSEs are less than 0015 Hence theresults of all estimates are satisfied
6 Conclusions
In this paper the Bayesian analysis for the step-stress partiallyaccelerated competing failuremodel fromPareto distributionis investigated under Type II progressive censoring Based onthe noninformative prior and informative prior distributionswe derive the objective Bayesian estimates and Bayesianestimates of the unknown parameters and acceleration factorrespectively Considering such case that the hyperparametersare unknown and follow certain distributions the E-Bayesianmethod is utilized to estimate the unknown parameters andacceleration factor We analyze two real data sets to illustratethe application of the Pareto distribution in engineering
practice Finally the comparisons of the efficiencies of theobjective Bayesian estimates Bayesian estimates and corre-sponding E-Bayesian estimates are conducted It is concludedthat the OBE performances are better than that of BE andE-BE2 performances are better than that of E-BE1 underthe different progressive censoring schemes In addition wediscuss the effects of sample sizes and different progressivecensoring schemes on the estimates The simulation resultsdemonstrate that for fixed n the AREs and MSEs of allestimates decrease with119898 increasing
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The work is supported by the National Natural ScienceFoundation of China (71401134 71571144 and 71171164)The Natural Science Basic Research Program of ShaanxiProvince (2015JM1003) and The International Cooperation
Mathematical Problems in Engineering 9
Table 3 The AREs and MSEs of all estimates for differentm n and for censoring Scheme 3
(119899 119898) Para OBEARE (MSE)
BEARE (MSE)
E-BE1ARE (MSE)
E-BE2ARE (MSE)
(40 10) 120582 00535 (00079) 00578 (00082) 00521 (00056) 00510 (00054)1205791 00358 (00053) 00385 (00083) 00399 (00082) 00390 (00072)1205792 00481 (00129) 00491 (00145) 00512 (00130) 00520 (00122)
(40 15) 120582 00523 (00072) 00551 (00080) 00520 (00052) 00501 (00051)1205791 00364 (00048) 00371 (00077) 00396 (00076) 00380 (00071)1205792 00471 (00122) 00480 (00142) 00511 (00129) 00512 (00120)
(40 20) 120582 00491 (00066) 00552 (00078) 00513 (00051) 00504 (00050)1205791 00342 (00043) 00350 (00075) 00381 (00071) 00373 (00068)1205792 00465 (00120) 00469 (00140) 00507 (00124) 00499 (00118)
(45 10) 120582 00532 (00075) 00571 (00079) 00521 (00052) 00504 (00049)1205791 00363 (00051) 00381 (00081) 00398 (00081) 00390 (00067)1205792 00480 (00127) 00491 (00141) 00512 (00128) 00520 (00120)
(45 15) 120582 00518 (00068) 00551 (00075) 00520 (00050) 00501 (00046)1205791 00358 (00050) 00371 (00075) 00391 (00076) 00381 (00064)1205792 00467 (00118) 00478 (00137) 00510 (00123) 00510 (00117)
(45 20) 120582 00491 (00060) 00552 (00072) 00513 (00047) 00450 (00042)1205791 00340 (00048) 00345 (00069) 00381 (00065) 00371 (00062)1205792 00462 (00115) 00470 (00129) 00503 (00120) 00493 (00113)
(50 10) 120582 00530 (00074) 00571 (00067) 00520 (00050) 00502 (00045)1205791 00353 (00050) 00373 (00078) 00387 (00079) 00380 (00064)1205792 00468 (00124) 00483 (00140) 00502 (00124) 00510 (00119)
(50 15) 120582 00510 (00067) 00546 (00064) 00512 (00056) 00492 (00051)1205791 00348 (00042) 00359 (00075) 00382 (00076) 00368 (00062)1205792 00459 (00120) 00468 (00137) 00500 (00121) 00500 (00115)
(50 20) 120582 00481 (00065) 00541 (00062) 00503 (00053) 00491 (00050)1205791 00330 (00040) 00335 (00072) 00368 (00073) 00369 (00058)1205792 00454 (00116) 00462 (00134) 00497 (00118) 00485 (00113)
and Exchanges in Science and Technology Program ofShaanxi Province (2016KW-033)
References
[1] D Han and D Kundu ldquoInference for a step-stress model withcompeting risks for failure from the generalized exponentialdistribution under type-I censoringrdquo IEEE Transactions onReliability vol 64 no 1 pp 31ndash43 2015
[2] S Roy and C Mukhopadhyay ldquoMaximum likelihood analysisof multi-stress accelerated life test data of series systems withcompeting log-normal causes of failurerdquo Journal of Risk andReliability vol 229 no 2 pp 119ndash130 2015
[3] X P Zhang J Z Shang X Chen C H Zhang and Y S WangldquoStatistical inference of accelerated life testing with dependentcompeting failures based on copula theoryrdquo IEEE Transactionson Reliability vol 63 no 3 pp 764ndash780 2014
[4] Y-M Shi L Jin C Wei and H-B Yue ldquoConstant-stressaccelerated life test with competing risks under progressivetype-II hybrid censoringrdquo Advanced Materials Research vol712ndash715 pp 2080ndash2083 2013
[5] A Xu and Y Tang ldquoObjective Bayesian analysis of acceleratedcompeting failure models under Type-I censoringrdquo Computa-tional Statistics andData Analysis vol 55 no 10 pp 2830ndash28392011
[6] MWu Y Shi and Y Sun ldquoInference for accelerated competingfailure models fromWeibull distribution under Type-I progres-sive hybrid censoringrdquo Journal of Computational and AppliedMathematics vol 263 pp 423ndash431 2014
[7] N Balakrishnan andD Han ldquoExact inference for a simple step-stress model with competing risks for failure from exponentialdistribution under type-II censoringrdquo Journal of StatisticalPlanning and Inference vol 138 no 12 pp 4172ndash4186 2008
[8] D Han and N Balakrishnan ldquoInference for a simple step-stressmodel with competing risks for failure from the exponentialdistribution under time constraintrdquo Computational Statisticsand Data Analysis vol 54 no 9 pp 2066ndash2081 2010
[9] A M Abd-Elfattah A S Hassan and S G Nassr ldquoEstimationin step-stress partially accelerated life tests for the Burr typeXII distribution using type I censoringrdquo StatisticalMethodologyvol 5 no 6 pp 502ndash514 2008
[10] A A Ismail ldquoBayes estimation of Gompertz distributionparameters and acceleration factor under partially accelerated
10 Mathematical Problems in Engineering
life tests with type-I censoringrdquo Journal of Statistical Computa-tion and Simulation vol 80 no 11 pp 1253ndash1264 2010
[11] A A Ismail ldquoEstimating the parameters ofWeibull distributionand the acceleration factor from hybrid partially accelerated lifetestrdquo Applied Mathematical Modelling vol 36 no 7 pp 2920ndash2925 2012
[12] T A Abushal and A A Soliman ldquoEstimating the Paretoparameters under progressive censoring data for constant-partially accelerated life testsrdquo Journal of Statistical Computationand Simulation vol 85 no 5 pp 917ndash934 2015
[13] A A Ismail and A M Sarhan ldquoOptimal design of step-stresslife test with progressively type-II censored exponential datardquoInternational Mathematical Forum vol 4 no 37ndash40 pp 1963ndash1976 2009
[14] H Jeffreys The Theory of Probability Oxford University PressLondon UK 1998
[15] Q Guan Y Tang and A Xu ldquoObjective Bayesian analysis forbivariate Marshall-Olkin exponential distributionrdquo Computa-tional Statistics and Data Analysis vol 64 pp 299ndash313 2013
[16] M Han ldquoE-Bayesian estimation of failure probability and itsapplicationrdquoMathematical and ComputerModelling vol 45 no9-10 pp 1272ndash1279 2007
[17] M Han ldquoE-Bayesian estimation and hierarchical Bayesian esti-mation of failure probabilityrdquo Communications in StatisticsmdashTheory and Methods vol 40 no 18 pp 3303ndash3314 2011
[18] N L Johnson S Kotz and N Balakrishnan ContinuousUnivariate Distributions vol 1 Wiley-Interscience New YorkNY USA 2nd edition 1994
[19] AA Soliman ldquoEstimations for Paretomodel using general pro-gressive censored data and asymmetric lossrdquo Communicationsin StatisticsTheory andMethods vol 37 no 8ndash10 pp 1353ndash13702008
[20] H T Davis and M L Feldstein ldquoThe generalized Pareto law asa model for progressively censored survival datardquo Biometrikavol 66 no 2 pp 299ndash306 1979
[21] C M Harris ldquoThe Pareto distribution as a queue servicedisciplinerdquoOperations Research vol 16 no 2 pp 307ndash313 1968
[22] M BourguignonH Saulo andRN Fernandez ldquoA newPareto-type distribution with applications in reliability and incomedatardquo Physica A Statistical Mechanics and Its Applications vol457 pp 166ndash175 2016
[23] M Doostparast M G Akbari and N Balakrishnan ldquoBayesiananalysis for the two-parameter Pareto distribution based onrecord values and timesrdquo Journal of Statistical Computation andSimulation vol 81 no 11 pp 1393ndash1403 2011
[24] J O Berger Statistical Decision Theory and Bayesian AnalysisSpringer New York NY USA 2nd edition 1985
[25] D N P Murthy M Xie and R Jiang ldquoWeibull modelsrdquo inWiley Series in Probability and Statistics John Wiley and SonsHoboken NJ USA 2004
[26] W B Nelson ldquoStatistical methods for accelerated life test datathe inverse power law modelrdquo General Electric Co Tech Rep71-C011 Schenectady New York NY USA 1970
[27] N Balakrishnan and R A Sandhu ldquoA simple simulationalalgorithm for generating progressive Type-II censored samplesrdquoThe American Statistician vol 49 no 2 pp 229ndash230 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
Therefore under the squared loss function the Bayesianestimate of 120582 can be approximated as
BS = 119867() + 1198671015840 () (1 + 119868minus11 minus )ℎ () + ℎ1015840 () (1 + 119868minus11 minus ) (19)
where 1198681 = (1205721 + 1205722)(ln1198821 minus ln1198822) + 11988733 E-Bayesian Estimates By assuming that the hyperparam-eters follow certain distributions the Bayesian estimates canbe obtained when the hyperparameters in prior distributionsare unknown However complicated integration is often ahard work Han [16 17] gave the definition of E-Bayesianestimation to overcome this problem So we will derivethe E-Bayesian estimates of the unknown parameters andacceleration factor in this subsection
Suppose that the hyperparameters 119886 and 119887 and 120573119895 and119889119895 (119895 = 1 2) have the joint prior density functions 120587119890(119886 119887)and 120587119890(120573119895 119889119895) respectively Then the E-Bayesian estimates of120582 and 120579119895 (119895 = 1 2) are
EB = ∬119863BS120587119890 (119886 119887) 119889119886 119889119887 (20)
119895EB = ∬119863119895BS120587119890 (120573119895 119889119895) 119889120573119895119889 (119889119895) (21)
where BS and 119895BS are the Bayesian estimates of 120582 and 120579119895 (119895 =1 2) respectivelyAccording to Han [16] 119886 and 119887 and 120573119895 and 119889119895 (119895 = 1 2)
should be selected to guarantee that 120587119904(120582) and 120587119904(120579119895) aredecreasing functions of 120582 and 120579119895 respectivelyThe derivativesof 120587119904(120582) with respect to 120582 and 120587119904(120579119895) with respect to 120579119895 are
120597120587119904 (120582)120597120582 prop 120582119886minus2 exp minus119887120582 (119886 minus 1 minus 119887120582) 120597120587119904 (120579119895)120597120579119895 = 120573119895 (120573119895 minus 1) 120579120573119895minus2119895 119889minus120573119895119895
(22)
When 0 lt 119886 lt 1 119887 gt 0 120597120587119904(120582)120597120582 lt 0 and when 0 lt 120573119895 lt 1119889119895 gt 0 120597120587119904(120579119895)120597120579119895 lt 0Thus120587119904(120582) and120587119904(120579119895) are decreasingfunctions of 120582 and 120579119895 respectively
Given that 0 lt 119886 lt 1 and 0 lt 120573119895 lt 1 thelarger 119887 and 119889119895 are the thinner the tail of 120587119904(120582) and 120587119904(120579119895)will be Berger [24] showed that the thinner tailed priordistribution often reduces the robustness of the Bayesianestimate Consequently 119887 and 119889119895 (119895 = 1 2) should begiven upper bounds 119897 and ℎ1015840119895 respectively We consider thefollowing two joint prior distributions of 119886 and 119887 and 120573119895 and119889119895 (119895 = 1 2) respectively
1205871 (119886 119887) = 21198871198972 0 lt 119886 lt 1 0 lt 119887 lt 1198971205872 (119886 119887) = 1119897 0 lt 119886 lt 1 0 lt 119887 lt 119897 (23)
1205871119895 (120573119895 119889119895) = 2119889119895ℎ10158402119895 0 lt 120573119895 lt 1 0 lt 119889119895 lt ℎ10158401198951205872119895 (120573119895 119889119895) = 1ℎ1015840119895 0 lt 120573119895 lt 1 0 lt 119889119895 lt ℎ1015840119895
(24)
From (19) (20) and (23) we derive the E-Bayesian estimatesof 120582 as
EB1 = 21198972 int119897
0int10
119887 [119867 () + 1198671015840 () (1 + 119868minus12 minus )]ℎ () + ℎ1015840 () (1 + 119868minus12 minus ) 119889119886 119889119887
= 21198972 int119897
0119887int10
(1198681 + 119887) + (119898 minus 1198731 + 119886) [1 + (1198681 + 119887) (1 minus )](1198681 + 119887) + (119898 minus 1198731 + 119886 minus 1) [1 + (1198681 + 119887) (1 minus )]119889119886 119889119887 = +1198972 (1198761 minus 1198762)
EB2 = 1119897 int119897
0int10
119867() + 1198671015840 () (1 + 119868minus12 minus )ℎ () + ℎ1015840 () (1 + 119868minus12 minus ) 119889119886 119889119887
= 119897 int119897
0int10
(1198681 + 119887) + (119898 minus 1198731 + 119886) [1 + (1198681 + 119887) (1 minus )](1198681 + 119887) + (119898 minus 1198731 + 119886 minus 1) [1 + (1198681 + 119887) (1 minus )]119889119886 119889119887 = +119897 (1198641 minus 1198642)
(25)
where
1198761 = [[1198972 minus (1198681 + 1198732119911 ())
2]] ln [(1198681 + 119897) 119911 () + 1198732]
+ (1198681 + 1198732119911 ())2
ln (1198681119911 () + 1198732) + 1198732119897119911 ()
1198762 = [[1198972 minus (1198681 + 1198732 minus 1119901 (1006704120582) )
2]]
sdot ln [(1198681 + 119897) 119901 (1006704120582) + 1198732 minus 1] + (1198681 + 1198732 minus 1119901 (1006704120582) )2
6 Mathematical Problems in Engineering
sdot ln (1198681119901 (1006704120582) + 1198732 minus 1) + (1198732 minus 1) 119897119901 (1006704120582) 1198641 = (1198681 + 119897 + 1198732119911 ()) ln [(1198681 + 119897) 119911 () + 1198732]minus (1198681 + 1198732119911 ()) ln [1198681119911 () + 1198732]
1198642 = (1198681 + 119897 + 1198732 minus 1119901 () ) ln [(1198681 + 119897) 119901 () + 1198732 minus 1]
minus (1198681 + 1198732 minus 1119901 () ) ln [1198681119901 () + 1198732 minus 1] 119911 () = + 1198732 (1 minus ) 119901 () = + (1198732 minus 1) (1 minus ) 1198732 = 119898 minus 1198731
(26)Similarly from (17) (21) and (24) the E-Bayesian estimate of120579119895 (119895 = 1 2) can be obtained as
119895EB1 =
2ℎ10158401198953 (1 minus ln120572119895119899 + 2120572119895119899 + 1) 119889lowast119895 = 119889119895
119889lowast119895 (1 minus ln120572119895119899 + 2120572119895119899 + 1) 119889lowast119895 = 119889119895
119895EB2 =
ℎ10158401198952 (1 minus ln120572119895119899 + 2120572119895119899 + 1) 119889lowast119895 = 119889119895
119889lowast119895 (1 minus ln120572119895119899 + 2120572119895119899 + 1) 119889lowast119895 = 119889119895
(27)
4 Application of the Pareto Distribution
In this section we analyze two real data sets to illustrate theapplication of the Pareto distribution in engineering practiceThe first data set (Set I) represent the failure time of 20mechanical components These data are taken from Murthyet al [25] and have been used by Bourguignon et al [22]These data are 0067 0068 0076 0081 0084 0085 00850086 0089 0098 0098 0114 0114 0115 0121 01250131 0149 0160 and 0485 The second data set (Set II) isgiven by Nelson [26] which represents the failure times tobreakdown of a type of electronic insulating material subjectto a constant-voltage stress These data are 035 059 096099 169 197 207 258 271 290 367 399 535 1377 and2550
We check the validity for the Pareto distribution 119875(120572 120579)by using the Kolmogorov test The statistics of the Kol-mogorov test are119863119899 = sup119909|119865(119909)minus119865119899(119909)| = max1le119894le20|119865(119909119894)minus119865119899(119909119894)| where 119865(119909) is the distribution function of 119875(120572 120579)and 119865(119909) = 1 minus (120579119909)120572 119909 gt 0 and 119865119899(119909) is the empiricaldistribution functions constructed based on the above dataSet I and Set II
For the data Set I when the parameters 120572 = 2113 and120579 = 0067 the observation value of the statistic is 119899 =max1le119894le20|119865(119909119894)minus119865119899(119909119894)| = 01303 For 119899 = 20 we choose thevalue of confidence level120572 as120572 = 001 002 005 respectivelyBy using the table of critical value on the Kolmogorov test weget the critical value 119899120572 where 20005 = 029403 20002 =032266 and 20001 = 035241 Since 119899 lt 20005 119899 lt20002 and 119899 lt 20001 thus the fit of Pareto distribution119875(120572 120579) = 119875(2113 0067) to the above data Set I is reasonable
For data Set II when the parameters 120572 = 051 and120579 = 035 the observation value of the statistic is 119899 =max1le119894le20|119865(119909119894)minus119865119899(119909119894)| = 02187 For 119899 = 15 we choose thevalue of confidence level120572 as120572 = 001 002 005 respectivelyBy using the table of critical value on the Kolmogorov testwe get the critical value 119899120572 where 15005 = 03376015002 = 037713 and 15001 = 040120 Since 119899 lt15005 119899 lt 15002 and 119899 lt 15001 the fit of Paretodistribution 119875(120572 120579) = 119875(051 035) to the above data Set IIis reasonable Therefore the use of the Pareto distribution iswell justified
5 Numerical Example
In this section we present numerical example using MonteCarlo simulations to illustrate the estimation procedurepresented in this paper The performance of three types ofBayesian estimates for different sample sizes and differentprogressive censoring schemes (PCS) is assessed based onsimulation dataThe term ldquodifferent PCSrdquomeans different setsof 119877119894rsquos In this paper we consider the following three differentPCS
Scheme 1 1198771 = sdot sdot sdot = 119877119898minus1 = 0 and 119877119898 = 119899 minus 119898Scheme 2 1198771 = 1198772 = sdot sdot sdot = 119877119871 = 0 119877119871+1 = sdot sdot sdot = 119877119898minus1 = 1and 119877119898 = 119899 minus 2119898 + 1 + 119871Scheme 3 1198771 = 1198772 = sdot sdot sdot = 119877119871+1 = 1 119877119871+2 = sdot sdot sdot = 119877119898minus1 =0 and 119877119898 = 119899 minus 119898 minus 1 minus 119871 where 119871 = [1198984] + 1 [1198984]representing the maximum integer that does not exceed1198984119898 ge 10
Before progressing further we first describe how togenerate Type II progressively censored competing failuredata from Pareto distribution under SSPALT
Step 1 Given119898 119899 and a certain censoring scheme generatetwo samples 1198801 1198802 119880119898 and 11988211198822 119882119898 from theuniform (0 1) distribution [27]
Step 2 Given 120572119895 120579119895 (119895 = 1 2) compute 11990511119894 and 11990512119894 bysolving equations 119880119894 = 1 minus 12057912057211 119905minus120572111119894 and 119882119894 = 1 minus 12057912057222 119905minus120572212119894 respectivelyThen record the minimum of (11990511119894 11990512119894) as 119905lowast119894 andthe corresponding index of the minimum (if 11990511119894 lt 11990512119894 set119888lowast119894 = 1 else set 119888lowast119894 = 2) for 1 le 119894 le 119898Step 3 For prefixed stress changing time 120591 find1198731 such that119905lowast1198731 le 120591 lt 119905lowast1198731+1 Then let 119905119894 = 119905lowast119894 and 119888119894 = 119888lowast119894 for 1 le 119894 le 1198731
Mathematical Problems in Engineering 7
Table 1 The AREs and MSEs of all estimates for differentm n and for censoring Scheme 1
(119899119898) Para OBEARE (MSE)
BEARE (MSE)
E-BE1ARE (MSE)
E-BE2ARE (MSE)
(40 10) 120582 00526 (00073) 00570 (00081) 00519 (00052) 00498 (0050)1205791 00360 (00048) 00378 (00076) 00392 (00075) 00382 (00065)1205792 00474 (00115) 00485 (00138) 00512 (00128) 00508 (00107)
(40 15) 120582 00514 (00069) 00551 (00067) 00515 (00048) 00495 (00044)1205791 00353 (00042) 00364 (00065) 00387 (00062) 00374 (00061)1205792 00462 (00107) 00472 (00120) 00504 (00112) 00500 (00099)
(40 20) 120582 00489 (00057) 00548 (00059) 00505 (00044) 00495 (00041)1205791 00336 (00032) 00341 (00056) 00372 (00051) 00365 (00049)1205792 00458 (00104) 00463 (00113) 00500 (00110) 00491 (00095)
(45 10) 120582 00513 (00070) 00564 (00080) 00514 (00050) 00498 (00048)1205791 00357 (00046) 00375 (00074) 00390 (00071) 00383 (00063)1205792 00471 (00111) 00486 (00127) 00505 (00115) 00512 (00100)
(45 15) 120582 00510 (00064) 00549 (00058) 00514 (00047) 00491 (00046)1205791 00350 (00040) 00362 (00061) 00386 (00059) 00370 (00060)1205792 00460 (00101) 00470 (00115) 00501 (00110) 00500 (00098)
(45 20) 120582 00484 (00056) 00542 (00054) 00504 (00041) 00493 (00038)1205791 00334 (00030) 00338 (00051) 00370 (00048) 00362 (00048)1205792 00454 (00100) 00462 (00111) 00497 (00105) 00489 (00087)
(50 10) 120582 00511 (00068) 00563 (00076) 00511 (00047) 00494 (00046)1205791 00353 (00044) 00373 (00071) 00387 (00070) 00380 (00061)1205792 00468 (00109) 00483 (00118) 00502 (00112) 00510 (00094)
(50 15) 120582 00509 (00062) 00546 (00058) 00512 (00045) 00492 (00045)1205791 00348 (00037) 00359 (00060) 00382 (00058) 00368 (00058)1205792 00459 (00095) 00468 (00110) 00500 (00107) 00500 (00093)
(50 20) 120582 00481 (00054) 00541 (00051) 00503 (00040) 00491 (00035)1205791 00330 (00027) 00335 (00047) 00368 (00042) 00369 (00043)1205792 00445 (00096) 00462 (00106) 00497 (00102) 00488 (00072)
Step 4 For 1198731 + 1 le 119894 le 119898 generate 11990521119894 by solvingthe equation 119880119894 = 1 minus 12057912057211 1205911205721(120582minus1)119905minus120572112058221119894 and generate 11990522119894 bysolving119882119894 = 1 minus 12057912057222 1205911205722(120582minus1)119905minus120572212058222119894 Again take the minimumof (11990521119894 11990522119894) as 119905lowast119894 as well as the corresponding index of theminimum (if 11990521119894 lt 11990522119894 set 119888lowast119894 = 1 else set 119888lowast119894 = 2)Step 5 For1198731 + 1 le 119894 le 119898 let 119905119894 = 119905lowast119894 and 119888119894 = 119888lowast119894
Then (1199051 1198881 1198771) (119905119898 119888119898 119877119898) is the required sample
We assumed that the occurrence time of the jth (119895 = 1 2)failure cause of tested product follows Pareto distributions119875(120572 120579) = 119875(2113 0067) and 119875(120572 120579) = 119875(182 0051) underuse stress levels 1198780 and 120582 = 20 and 120591 = 011 the parametersof prior density are taken as 119886 = 08 119887 = 20 1205731 = 0851205732 = 06 1198891 = 18 and 1198892 = 21 Considering different119898 119899 and censoring schemes we generate samples accordingto the above steps We replicate the process of evaluating1000 times in each case and compute the average relativeerrors (AREs) and mean squared errors (MSEs) ARE andMSE of the estimation of the parameter 120579 can be calculatedaccording to the following formulas
ARE () = 110001000sum119896=1
1003816100381610038161003816100381610038161003816120579 minus (119896)1003816100381610038161003816100381610038161003816120579
MSE () = 110001000sum119896=1
(120579 minus (119896))2 (28)
where (119896) is the 119896th estimator for the parameter 120579We calculate the AREs and MSEs of the objective
Bayesian estimates (OBE) Bayesian estimates (BE) andE-Bayesian estimates (E-BE) respectively The simulationresults are presented in Tables 1ndash3 In Tables 1ndash3 we denotethe E-BE based on the first prior as E-BE1 and the E-BE basedon the second prior as E-BE2
FromTables 1ndash3 the following observations can bemade
(1) For fixed n as m increases the AREs and MSEsdecrease for all estimates
(2) In terms of the values of AREs and MSEs the OBEperformances are better than that of BE and E-BE2
8 Mathematical Problems in Engineering
Table 2 The AREs and MSEs of all estimates for differentm n and for censoring Scheme 2
(119899 119898) Para OBEARE (MSE)
BEARE (MSE)
E-BE1ARE (MSE)
E-BE2ARE (MSE)
(40 10) 120582 00531 (00080) 00576 (00084) 00521 (00055) 00501 (00053)1205791 00365 (00051) 00381 (00081) 00398 (00079) 00386 (00067)1205792 00478 (00118) 00488 (00142) 00511 (00125) 00516 (00117)
(40 15) 120582 00516 (00071) 00551 (00082) 00515 (00051) 00495 (00049)1205791 00357 (00046) 00364 (00080) 00387 (00064) 00374 (00061)1205792 00465 (00113) 00472 (00138) 00504 (00114) 00500 (00112)
(40 20) 120582 00489 (00077) 00548 (00078) 00505 (00058) 00495 (00056)1205791 00336 (00044) 00341 (00075) 00372 (00056) 00365 (00053)1205792 00458 (00110) 00463 (00133) 00500 (00112) 00491 (00110)
(45 10) 120582 00527 (00079) 00564 (00079) 00511 (00052) 00501 (00051)1205791 00365 (00048) 00380 (00077) 00394 (00075) 00382 (00065)1205792 00474 (00115) 00485 (00133) 00509 (00122) 00512 (00114)
(45 15) 120582 00513 (00067) 00549 (00075) 00513 (00050) 00492 (00048)1205791 00348 (00037) 00364 (00074) 00382 (00062) 00370 (00058)1205792 00464 (00112) 00470 (00131) 00501 (00108) 00487 (00107)
(45 20) 120582 00486 (00065) 00546 (00071) 00501 (00057) 00492 (00045)1205791 00332 (00034) 00341 (00070) 00372 (00056) 00365 (00051)1205792 00451 (00110) 00457 (00127) 00492 (00104) 00490 (00104)
(50 10) 120582 00521 (00075) 00560 (00076) 00508 (00046) 00500 (00043)1205791 00362 (00046) 00380 (00075) 00392 (00074) 00381 (00063)1205792 00471 (00113) 00483 (00131) 00504 (00120) 00510 (00112)
(50 15) 120582 00504 (00065) 00546 (00074) 00512 (00050) 00491 (00045)1205791 00354 (00036) 00364 (00072) 00382 (00061) 00370 (00055)1205792 00464 (00110) 00470 (00130) 00501 (00105) 00496 (00104)
(50 20) 120582 00475 (00059) 00536 (00072) 00491 (00048) 00487 (00044)1205791 00324 (00029) 00332 (00170) 00366 (00052) 00356 (00051)1205792 00443 (00107) 00450 (00127) 00483 (00102) 00481 (00100)
performances are better than that of E-BE1 under thedifferent progressive censoring schemes
(3) Under the different progressive censoring schemesfor all estimates the values of AREs are less than 006and the values of MSEs are less than 0015 Hence theresults of all estimates are satisfied
6 Conclusions
In this paper the Bayesian analysis for the step-stress partiallyaccelerated competing failuremodel fromPareto distributionis investigated under Type II progressive censoring Based onthe noninformative prior and informative prior distributionswe derive the objective Bayesian estimates and Bayesianestimates of the unknown parameters and acceleration factorrespectively Considering such case that the hyperparametersare unknown and follow certain distributions the E-Bayesianmethod is utilized to estimate the unknown parameters andacceleration factor We analyze two real data sets to illustratethe application of the Pareto distribution in engineering
practice Finally the comparisons of the efficiencies of theobjective Bayesian estimates Bayesian estimates and corre-sponding E-Bayesian estimates are conducted It is concludedthat the OBE performances are better than that of BE andE-BE2 performances are better than that of E-BE1 underthe different progressive censoring schemes In addition wediscuss the effects of sample sizes and different progressivecensoring schemes on the estimates The simulation resultsdemonstrate that for fixed n the AREs and MSEs of allestimates decrease with119898 increasing
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The work is supported by the National Natural ScienceFoundation of China (71401134 71571144 and 71171164)The Natural Science Basic Research Program of ShaanxiProvince (2015JM1003) and The International Cooperation
Mathematical Problems in Engineering 9
Table 3 The AREs and MSEs of all estimates for differentm n and for censoring Scheme 3
(119899 119898) Para OBEARE (MSE)
BEARE (MSE)
E-BE1ARE (MSE)
E-BE2ARE (MSE)
(40 10) 120582 00535 (00079) 00578 (00082) 00521 (00056) 00510 (00054)1205791 00358 (00053) 00385 (00083) 00399 (00082) 00390 (00072)1205792 00481 (00129) 00491 (00145) 00512 (00130) 00520 (00122)
(40 15) 120582 00523 (00072) 00551 (00080) 00520 (00052) 00501 (00051)1205791 00364 (00048) 00371 (00077) 00396 (00076) 00380 (00071)1205792 00471 (00122) 00480 (00142) 00511 (00129) 00512 (00120)
(40 20) 120582 00491 (00066) 00552 (00078) 00513 (00051) 00504 (00050)1205791 00342 (00043) 00350 (00075) 00381 (00071) 00373 (00068)1205792 00465 (00120) 00469 (00140) 00507 (00124) 00499 (00118)
(45 10) 120582 00532 (00075) 00571 (00079) 00521 (00052) 00504 (00049)1205791 00363 (00051) 00381 (00081) 00398 (00081) 00390 (00067)1205792 00480 (00127) 00491 (00141) 00512 (00128) 00520 (00120)
(45 15) 120582 00518 (00068) 00551 (00075) 00520 (00050) 00501 (00046)1205791 00358 (00050) 00371 (00075) 00391 (00076) 00381 (00064)1205792 00467 (00118) 00478 (00137) 00510 (00123) 00510 (00117)
(45 20) 120582 00491 (00060) 00552 (00072) 00513 (00047) 00450 (00042)1205791 00340 (00048) 00345 (00069) 00381 (00065) 00371 (00062)1205792 00462 (00115) 00470 (00129) 00503 (00120) 00493 (00113)
(50 10) 120582 00530 (00074) 00571 (00067) 00520 (00050) 00502 (00045)1205791 00353 (00050) 00373 (00078) 00387 (00079) 00380 (00064)1205792 00468 (00124) 00483 (00140) 00502 (00124) 00510 (00119)
(50 15) 120582 00510 (00067) 00546 (00064) 00512 (00056) 00492 (00051)1205791 00348 (00042) 00359 (00075) 00382 (00076) 00368 (00062)1205792 00459 (00120) 00468 (00137) 00500 (00121) 00500 (00115)
(50 20) 120582 00481 (00065) 00541 (00062) 00503 (00053) 00491 (00050)1205791 00330 (00040) 00335 (00072) 00368 (00073) 00369 (00058)1205792 00454 (00116) 00462 (00134) 00497 (00118) 00485 (00113)
and Exchanges in Science and Technology Program ofShaanxi Province (2016KW-033)
References
[1] D Han and D Kundu ldquoInference for a step-stress model withcompeting risks for failure from the generalized exponentialdistribution under type-I censoringrdquo IEEE Transactions onReliability vol 64 no 1 pp 31ndash43 2015
[2] S Roy and C Mukhopadhyay ldquoMaximum likelihood analysisof multi-stress accelerated life test data of series systems withcompeting log-normal causes of failurerdquo Journal of Risk andReliability vol 229 no 2 pp 119ndash130 2015
[3] X P Zhang J Z Shang X Chen C H Zhang and Y S WangldquoStatistical inference of accelerated life testing with dependentcompeting failures based on copula theoryrdquo IEEE Transactionson Reliability vol 63 no 3 pp 764ndash780 2014
[4] Y-M Shi L Jin C Wei and H-B Yue ldquoConstant-stressaccelerated life test with competing risks under progressivetype-II hybrid censoringrdquo Advanced Materials Research vol712ndash715 pp 2080ndash2083 2013
[5] A Xu and Y Tang ldquoObjective Bayesian analysis of acceleratedcompeting failure models under Type-I censoringrdquo Computa-tional Statistics andData Analysis vol 55 no 10 pp 2830ndash28392011
[6] MWu Y Shi and Y Sun ldquoInference for accelerated competingfailure models fromWeibull distribution under Type-I progres-sive hybrid censoringrdquo Journal of Computational and AppliedMathematics vol 263 pp 423ndash431 2014
[7] N Balakrishnan andD Han ldquoExact inference for a simple step-stress model with competing risks for failure from exponentialdistribution under type-II censoringrdquo Journal of StatisticalPlanning and Inference vol 138 no 12 pp 4172ndash4186 2008
[8] D Han and N Balakrishnan ldquoInference for a simple step-stressmodel with competing risks for failure from the exponentialdistribution under time constraintrdquo Computational Statisticsand Data Analysis vol 54 no 9 pp 2066ndash2081 2010
[9] A M Abd-Elfattah A S Hassan and S G Nassr ldquoEstimationin step-stress partially accelerated life tests for the Burr typeXII distribution using type I censoringrdquo StatisticalMethodologyvol 5 no 6 pp 502ndash514 2008
[10] A A Ismail ldquoBayes estimation of Gompertz distributionparameters and acceleration factor under partially accelerated
10 Mathematical Problems in Engineering
life tests with type-I censoringrdquo Journal of Statistical Computa-tion and Simulation vol 80 no 11 pp 1253ndash1264 2010
[11] A A Ismail ldquoEstimating the parameters ofWeibull distributionand the acceleration factor from hybrid partially accelerated lifetestrdquo Applied Mathematical Modelling vol 36 no 7 pp 2920ndash2925 2012
[12] T A Abushal and A A Soliman ldquoEstimating the Paretoparameters under progressive censoring data for constant-partially accelerated life testsrdquo Journal of Statistical Computationand Simulation vol 85 no 5 pp 917ndash934 2015
[13] A A Ismail and A M Sarhan ldquoOptimal design of step-stresslife test with progressively type-II censored exponential datardquoInternational Mathematical Forum vol 4 no 37ndash40 pp 1963ndash1976 2009
[14] H Jeffreys The Theory of Probability Oxford University PressLondon UK 1998
[15] Q Guan Y Tang and A Xu ldquoObjective Bayesian analysis forbivariate Marshall-Olkin exponential distributionrdquo Computa-tional Statistics and Data Analysis vol 64 pp 299ndash313 2013
[16] M Han ldquoE-Bayesian estimation of failure probability and itsapplicationrdquoMathematical and ComputerModelling vol 45 no9-10 pp 1272ndash1279 2007
[17] M Han ldquoE-Bayesian estimation and hierarchical Bayesian esti-mation of failure probabilityrdquo Communications in StatisticsmdashTheory and Methods vol 40 no 18 pp 3303ndash3314 2011
[18] N L Johnson S Kotz and N Balakrishnan ContinuousUnivariate Distributions vol 1 Wiley-Interscience New YorkNY USA 2nd edition 1994
[19] AA Soliman ldquoEstimations for Paretomodel using general pro-gressive censored data and asymmetric lossrdquo Communicationsin StatisticsTheory andMethods vol 37 no 8ndash10 pp 1353ndash13702008
[20] H T Davis and M L Feldstein ldquoThe generalized Pareto law asa model for progressively censored survival datardquo Biometrikavol 66 no 2 pp 299ndash306 1979
[21] C M Harris ldquoThe Pareto distribution as a queue servicedisciplinerdquoOperations Research vol 16 no 2 pp 307ndash313 1968
[22] M BourguignonH Saulo andRN Fernandez ldquoA newPareto-type distribution with applications in reliability and incomedatardquo Physica A Statistical Mechanics and Its Applications vol457 pp 166ndash175 2016
[23] M Doostparast M G Akbari and N Balakrishnan ldquoBayesiananalysis for the two-parameter Pareto distribution based onrecord values and timesrdquo Journal of Statistical Computation andSimulation vol 81 no 11 pp 1393ndash1403 2011
[24] J O Berger Statistical Decision Theory and Bayesian AnalysisSpringer New York NY USA 2nd edition 1985
[25] D N P Murthy M Xie and R Jiang ldquoWeibull modelsrdquo inWiley Series in Probability and Statistics John Wiley and SonsHoboken NJ USA 2004
[26] W B Nelson ldquoStatistical methods for accelerated life test datathe inverse power law modelrdquo General Electric Co Tech Rep71-C011 Schenectady New York NY USA 1970
[27] N Balakrishnan and R A Sandhu ldquoA simple simulationalalgorithm for generating progressive Type-II censored samplesrdquoThe American Statistician vol 49 no 2 pp 229ndash230 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
sdot ln (1198681119901 (1006704120582) + 1198732 minus 1) + (1198732 minus 1) 119897119901 (1006704120582) 1198641 = (1198681 + 119897 + 1198732119911 ()) ln [(1198681 + 119897) 119911 () + 1198732]minus (1198681 + 1198732119911 ()) ln [1198681119911 () + 1198732]
1198642 = (1198681 + 119897 + 1198732 minus 1119901 () ) ln [(1198681 + 119897) 119901 () + 1198732 minus 1]
minus (1198681 + 1198732 minus 1119901 () ) ln [1198681119901 () + 1198732 minus 1] 119911 () = + 1198732 (1 minus ) 119901 () = + (1198732 minus 1) (1 minus ) 1198732 = 119898 minus 1198731
(26)Similarly from (17) (21) and (24) the E-Bayesian estimate of120579119895 (119895 = 1 2) can be obtained as
119895EB1 =
2ℎ10158401198953 (1 minus ln120572119895119899 + 2120572119895119899 + 1) 119889lowast119895 = 119889119895
119889lowast119895 (1 minus ln120572119895119899 + 2120572119895119899 + 1) 119889lowast119895 = 119889119895
119895EB2 =
ℎ10158401198952 (1 minus ln120572119895119899 + 2120572119895119899 + 1) 119889lowast119895 = 119889119895
119889lowast119895 (1 minus ln120572119895119899 + 2120572119895119899 + 1) 119889lowast119895 = 119889119895
(27)
4 Application of the Pareto Distribution
In this section we analyze two real data sets to illustrate theapplication of the Pareto distribution in engineering practiceThe first data set (Set I) represent the failure time of 20mechanical components These data are taken from Murthyet al [25] and have been used by Bourguignon et al [22]These data are 0067 0068 0076 0081 0084 0085 00850086 0089 0098 0098 0114 0114 0115 0121 01250131 0149 0160 and 0485 The second data set (Set II) isgiven by Nelson [26] which represents the failure times tobreakdown of a type of electronic insulating material subjectto a constant-voltage stress These data are 035 059 096099 169 197 207 258 271 290 367 399 535 1377 and2550
We check the validity for the Pareto distribution 119875(120572 120579)by using the Kolmogorov test The statistics of the Kol-mogorov test are119863119899 = sup119909|119865(119909)minus119865119899(119909)| = max1le119894le20|119865(119909119894)minus119865119899(119909119894)| where 119865(119909) is the distribution function of 119875(120572 120579)and 119865(119909) = 1 minus (120579119909)120572 119909 gt 0 and 119865119899(119909) is the empiricaldistribution functions constructed based on the above dataSet I and Set II
For the data Set I when the parameters 120572 = 2113 and120579 = 0067 the observation value of the statistic is 119899 =max1le119894le20|119865(119909119894)minus119865119899(119909119894)| = 01303 For 119899 = 20 we choose thevalue of confidence level120572 as120572 = 001 002 005 respectivelyBy using the table of critical value on the Kolmogorov test weget the critical value 119899120572 where 20005 = 029403 20002 =032266 and 20001 = 035241 Since 119899 lt 20005 119899 lt20002 and 119899 lt 20001 thus the fit of Pareto distribution119875(120572 120579) = 119875(2113 0067) to the above data Set I is reasonable
For data Set II when the parameters 120572 = 051 and120579 = 035 the observation value of the statistic is 119899 =max1le119894le20|119865(119909119894)minus119865119899(119909119894)| = 02187 For 119899 = 15 we choose thevalue of confidence level120572 as120572 = 001 002 005 respectivelyBy using the table of critical value on the Kolmogorov testwe get the critical value 119899120572 where 15005 = 03376015002 = 037713 and 15001 = 040120 Since 119899 lt15005 119899 lt 15002 and 119899 lt 15001 the fit of Paretodistribution 119875(120572 120579) = 119875(051 035) to the above data Set IIis reasonable Therefore the use of the Pareto distribution iswell justified
5 Numerical Example
In this section we present numerical example using MonteCarlo simulations to illustrate the estimation procedurepresented in this paper The performance of three types ofBayesian estimates for different sample sizes and differentprogressive censoring schemes (PCS) is assessed based onsimulation dataThe term ldquodifferent PCSrdquomeans different setsof 119877119894rsquos In this paper we consider the following three differentPCS
Scheme 1 1198771 = sdot sdot sdot = 119877119898minus1 = 0 and 119877119898 = 119899 minus 119898Scheme 2 1198771 = 1198772 = sdot sdot sdot = 119877119871 = 0 119877119871+1 = sdot sdot sdot = 119877119898minus1 = 1and 119877119898 = 119899 minus 2119898 + 1 + 119871Scheme 3 1198771 = 1198772 = sdot sdot sdot = 119877119871+1 = 1 119877119871+2 = sdot sdot sdot = 119877119898minus1 =0 and 119877119898 = 119899 minus 119898 minus 1 minus 119871 where 119871 = [1198984] + 1 [1198984]representing the maximum integer that does not exceed1198984119898 ge 10
Before progressing further we first describe how togenerate Type II progressively censored competing failuredata from Pareto distribution under SSPALT
Step 1 Given119898 119899 and a certain censoring scheme generatetwo samples 1198801 1198802 119880119898 and 11988211198822 119882119898 from theuniform (0 1) distribution [27]
Step 2 Given 120572119895 120579119895 (119895 = 1 2) compute 11990511119894 and 11990512119894 bysolving equations 119880119894 = 1 minus 12057912057211 119905minus120572111119894 and 119882119894 = 1 minus 12057912057222 119905minus120572212119894 respectivelyThen record the minimum of (11990511119894 11990512119894) as 119905lowast119894 andthe corresponding index of the minimum (if 11990511119894 lt 11990512119894 set119888lowast119894 = 1 else set 119888lowast119894 = 2) for 1 le 119894 le 119898Step 3 For prefixed stress changing time 120591 find1198731 such that119905lowast1198731 le 120591 lt 119905lowast1198731+1 Then let 119905119894 = 119905lowast119894 and 119888119894 = 119888lowast119894 for 1 le 119894 le 1198731
Mathematical Problems in Engineering 7
Table 1 The AREs and MSEs of all estimates for differentm n and for censoring Scheme 1
(119899119898) Para OBEARE (MSE)
BEARE (MSE)
E-BE1ARE (MSE)
E-BE2ARE (MSE)
(40 10) 120582 00526 (00073) 00570 (00081) 00519 (00052) 00498 (0050)1205791 00360 (00048) 00378 (00076) 00392 (00075) 00382 (00065)1205792 00474 (00115) 00485 (00138) 00512 (00128) 00508 (00107)
(40 15) 120582 00514 (00069) 00551 (00067) 00515 (00048) 00495 (00044)1205791 00353 (00042) 00364 (00065) 00387 (00062) 00374 (00061)1205792 00462 (00107) 00472 (00120) 00504 (00112) 00500 (00099)
(40 20) 120582 00489 (00057) 00548 (00059) 00505 (00044) 00495 (00041)1205791 00336 (00032) 00341 (00056) 00372 (00051) 00365 (00049)1205792 00458 (00104) 00463 (00113) 00500 (00110) 00491 (00095)
(45 10) 120582 00513 (00070) 00564 (00080) 00514 (00050) 00498 (00048)1205791 00357 (00046) 00375 (00074) 00390 (00071) 00383 (00063)1205792 00471 (00111) 00486 (00127) 00505 (00115) 00512 (00100)
(45 15) 120582 00510 (00064) 00549 (00058) 00514 (00047) 00491 (00046)1205791 00350 (00040) 00362 (00061) 00386 (00059) 00370 (00060)1205792 00460 (00101) 00470 (00115) 00501 (00110) 00500 (00098)
(45 20) 120582 00484 (00056) 00542 (00054) 00504 (00041) 00493 (00038)1205791 00334 (00030) 00338 (00051) 00370 (00048) 00362 (00048)1205792 00454 (00100) 00462 (00111) 00497 (00105) 00489 (00087)
(50 10) 120582 00511 (00068) 00563 (00076) 00511 (00047) 00494 (00046)1205791 00353 (00044) 00373 (00071) 00387 (00070) 00380 (00061)1205792 00468 (00109) 00483 (00118) 00502 (00112) 00510 (00094)
(50 15) 120582 00509 (00062) 00546 (00058) 00512 (00045) 00492 (00045)1205791 00348 (00037) 00359 (00060) 00382 (00058) 00368 (00058)1205792 00459 (00095) 00468 (00110) 00500 (00107) 00500 (00093)
(50 20) 120582 00481 (00054) 00541 (00051) 00503 (00040) 00491 (00035)1205791 00330 (00027) 00335 (00047) 00368 (00042) 00369 (00043)1205792 00445 (00096) 00462 (00106) 00497 (00102) 00488 (00072)
Step 4 For 1198731 + 1 le 119894 le 119898 generate 11990521119894 by solvingthe equation 119880119894 = 1 minus 12057912057211 1205911205721(120582minus1)119905minus120572112058221119894 and generate 11990522119894 bysolving119882119894 = 1 minus 12057912057222 1205911205722(120582minus1)119905minus120572212058222119894 Again take the minimumof (11990521119894 11990522119894) as 119905lowast119894 as well as the corresponding index of theminimum (if 11990521119894 lt 11990522119894 set 119888lowast119894 = 1 else set 119888lowast119894 = 2)Step 5 For1198731 + 1 le 119894 le 119898 let 119905119894 = 119905lowast119894 and 119888119894 = 119888lowast119894
Then (1199051 1198881 1198771) (119905119898 119888119898 119877119898) is the required sample
We assumed that the occurrence time of the jth (119895 = 1 2)failure cause of tested product follows Pareto distributions119875(120572 120579) = 119875(2113 0067) and 119875(120572 120579) = 119875(182 0051) underuse stress levels 1198780 and 120582 = 20 and 120591 = 011 the parametersof prior density are taken as 119886 = 08 119887 = 20 1205731 = 0851205732 = 06 1198891 = 18 and 1198892 = 21 Considering different119898 119899 and censoring schemes we generate samples accordingto the above steps We replicate the process of evaluating1000 times in each case and compute the average relativeerrors (AREs) and mean squared errors (MSEs) ARE andMSE of the estimation of the parameter 120579 can be calculatedaccording to the following formulas
ARE () = 110001000sum119896=1
1003816100381610038161003816100381610038161003816120579 minus (119896)1003816100381610038161003816100381610038161003816120579
MSE () = 110001000sum119896=1
(120579 minus (119896))2 (28)
where (119896) is the 119896th estimator for the parameter 120579We calculate the AREs and MSEs of the objective
Bayesian estimates (OBE) Bayesian estimates (BE) andE-Bayesian estimates (E-BE) respectively The simulationresults are presented in Tables 1ndash3 In Tables 1ndash3 we denotethe E-BE based on the first prior as E-BE1 and the E-BE basedon the second prior as E-BE2
FromTables 1ndash3 the following observations can bemade
(1) For fixed n as m increases the AREs and MSEsdecrease for all estimates
(2) In terms of the values of AREs and MSEs the OBEperformances are better than that of BE and E-BE2
8 Mathematical Problems in Engineering
Table 2 The AREs and MSEs of all estimates for differentm n and for censoring Scheme 2
(119899 119898) Para OBEARE (MSE)
BEARE (MSE)
E-BE1ARE (MSE)
E-BE2ARE (MSE)
(40 10) 120582 00531 (00080) 00576 (00084) 00521 (00055) 00501 (00053)1205791 00365 (00051) 00381 (00081) 00398 (00079) 00386 (00067)1205792 00478 (00118) 00488 (00142) 00511 (00125) 00516 (00117)
(40 15) 120582 00516 (00071) 00551 (00082) 00515 (00051) 00495 (00049)1205791 00357 (00046) 00364 (00080) 00387 (00064) 00374 (00061)1205792 00465 (00113) 00472 (00138) 00504 (00114) 00500 (00112)
(40 20) 120582 00489 (00077) 00548 (00078) 00505 (00058) 00495 (00056)1205791 00336 (00044) 00341 (00075) 00372 (00056) 00365 (00053)1205792 00458 (00110) 00463 (00133) 00500 (00112) 00491 (00110)
(45 10) 120582 00527 (00079) 00564 (00079) 00511 (00052) 00501 (00051)1205791 00365 (00048) 00380 (00077) 00394 (00075) 00382 (00065)1205792 00474 (00115) 00485 (00133) 00509 (00122) 00512 (00114)
(45 15) 120582 00513 (00067) 00549 (00075) 00513 (00050) 00492 (00048)1205791 00348 (00037) 00364 (00074) 00382 (00062) 00370 (00058)1205792 00464 (00112) 00470 (00131) 00501 (00108) 00487 (00107)
(45 20) 120582 00486 (00065) 00546 (00071) 00501 (00057) 00492 (00045)1205791 00332 (00034) 00341 (00070) 00372 (00056) 00365 (00051)1205792 00451 (00110) 00457 (00127) 00492 (00104) 00490 (00104)
(50 10) 120582 00521 (00075) 00560 (00076) 00508 (00046) 00500 (00043)1205791 00362 (00046) 00380 (00075) 00392 (00074) 00381 (00063)1205792 00471 (00113) 00483 (00131) 00504 (00120) 00510 (00112)
(50 15) 120582 00504 (00065) 00546 (00074) 00512 (00050) 00491 (00045)1205791 00354 (00036) 00364 (00072) 00382 (00061) 00370 (00055)1205792 00464 (00110) 00470 (00130) 00501 (00105) 00496 (00104)
(50 20) 120582 00475 (00059) 00536 (00072) 00491 (00048) 00487 (00044)1205791 00324 (00029) 00332 (00170) 00366 (00052) 00356 (00051)1205792 00443 (00107) 00450 (00127) 00483 (00102) 00481 (00100)
performances are better than that of E-BE1 under thedifferent progressive censoring schemes
(3) Under the different progressive censoring schemesfor all estimates the values of AREs are less than 006and the values of MSEs are less than 0015 Hence theresults of all estimates are satisfied
6 Conclusions
In this paper the Bayesian analysis for the step-stress partiallyaccelerated competing failuremodel fromPareto distributionis investigated under Type II progressive censoring Based onthe noninformative prior and informative prior distributionswe derive the objective Bayesian estimates and Bayesianestimates of the unknown parameters and acceleration factorrespectively Considering such case that the hyperparametersare unknown and follow certain distributions the E-Bayesianmethod is utilized to estimate the unknown parameters andacceleration factor We analyze two real data sets to illustratethe application of the Pareto distribution in engineering
practice Finally the comparisons of the efficiencies of theobjective Bayesian estimates Bayesian estimates and corre-sponding E-Bayesian estimates are conducted It is concludedthat the OBE performances are better than that of BE andE-BE2 performances are better than that of E-BE1 underthe different progressive censoring schemes In addition wediscuss the effects of sample sizes and different progressivecensoring schemes on the estimates The simulation resultsdemonstrate that for fixed n the AREs and MSEs of allestimates decrease with119898 increasing
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The work is supported by the National Natural ScienceFoundation of China (71401134 71571144 and 71171164)The Natural Science Basic Research Program of ShaanxiProvince (2015JM1003) and The International Cooperation
Mathematical Problems in Engineering 9
Table 3 The AREs and MSEs of all estimates for differentm n and for censoring Scheme 3
(119899 119898) Para OBEARE (MSE)
BEARE (MSE)
E-BE1ARE (MSE)
E-BE2ARE (MSE)
(40 10) 120582 00535 (00079) 00578 (00082) 00521 (00056) 00510 (00054)1205791 00358 (00053) 00385 (00083) 00399 (00082) 00390 (00072)1205792 00481 (00129) 00491 (00145) 00512 (00130) 00520 (00122)
(40 15) 120582 00523 (00072) 00551 (00080) 00520 (00052) 00501 (00051)1205791 00364 (00048) 00371 (00077) 00396 (00076) 00380 (00071)1205792 00471 (00122) 00480 (00142) 00511 (00129) 00512 (00120)
(40 20) 120582 00491 (00066) 00552 (00078) 00513 (00051) 00504 (00050)1205791 00342 (00043) 00350 (00075) 00381 (00071) 00373 (00068)1205792 00465 (00120) 00469 (00140) 00507 (00124) 00499 (00118)
(45 10) 120582 00532 (00075) 00571 (00079) 00521 (00052) 00504 (00049)1205791 00363 (00051) 00381 (00081) 00398 (00081) 00390 (00067)1205792 00480 (00127) 00491 (00141) 00512 (00128) 00520 (00120)
(45 15) 120582 00518 (00068) 00551 (00075) 00520 (00050) 00501 (00046)1205791 00358 (00050) 00371 (00075) 00391 (00076) 00381 (00064)1205792 00467 (00118) 00478 (00137) 00510 (00123) 00510 (00117)
(45 20) 120582 00491 (00060) 00552 (00072) 00513 (00047) 00450 (00042)1205791 00340 (00048) 00345 (00069) 00381 (00065) 00371 (00062)1205792 00462 (00115) 00470 (00129) 00503 (00120) 00493 (00113)
(50 10) 120582 00530 (00074) 00571 (00067) 00520 (00050) 00502 (00045)1205791 00353 (00050) 00373 (00078) 00387 (00079) 00380 (00064)1205792 00468 (00124) 00483 (00140) 00502 (00124) 00510 (00119)
(50 15) 120582 00510 (00067) 00546 (00064) 00512 (00056) 00492 (00051)1205791 00348 (00042) 00359 (00075) 00382 (00076) 00368 (00062)1205792 00459 (00120) 00468 (00137) 00500 (00121) 00500 (00115)
(50 20) 120582 00481 (00065) 00541 (00062) 00503 (00053) 00491 (00050)1205791 00330 (00040) 00335 (00072) 00368 (00073) 00369 (00058)1205792 00454 (00116) 00462 (00134) 00497 (00118) 00485 (00113)
and Exchanges in Science and Technology Program ofShaanxi Province (2016KW-033)
References
[1] D Han and D Kundu ldquoInference for a step-stress model withcompeting risks for failure from the generalized exponentialdistribution under type-I censoringrdquo IEEE Transactions onReliability vol 64 no 1 pp 31ndash43 2015
[2] S Roy and C Mukhopadhyay ldquoMaximum likelihood analysisof multi-stress accelerated life test data of series systems withcompeting log-normal causes of failurerdquo Journal of Risk andReliability vol 229 no 2 pp 119ndash130 2015
[3] X P Zhang J Z Shang X Chen C H Zhang and Y S WangldquoStatistical inference of accelerated life testing with dependentcompeting failures based on copula theoryrdquo IEEE Transactionson Reliability vol 63 no 3 pp 764ndash780 2014
[4] Y-M Shi L Jin C Wei and H-B Yue ldquoConstant-stressaccelerated life test with competing risks under progressivetype-II hybrid censoringrdquo Advanced Materials Research vol712ndash715 pp 2080ndash2083 2013
[5] A Xu and Y Tang ldquoObjective Bayesian analysis of acceleratedcompeting failure models under Type-I censoringrdquo Computa-tional Statistics andData Analysis vol 55 no 10 pp 2830ndash28392011
[6] MWu Y Shi and Y Sun ldquoInference for accelerated competingfailure models fromWeibull distribution under Type-I progres-sive hybrid censoringrdquo Journal of Computational and AppliedMathematics vol 263 pp 423ndash431 2014
[7] N Balakrishnan andD Han ldquoExact inference for a simple step-stress model with competing risks for failure from exponentialdistribution under type-II censoringrdquo Journal of StatisticalPlanning and Inference vol 138 no 12 pp 4172ndash4186 2008
[8] D Han and N Balakrishnan ldquoInference for a simple step-stressmodel with competing risks for failure from the exponentialdistribution under time constraintrdquo Computational Statisticsand Data Analysis vol 54 no 9 pp 2066ndash2081 2010
[9] A M Abd-Elfattah A S Hassan and S G Nassr ldquoEstimationin step-stress partially accelerated life tests for the Burr typeXII distribution using type I censoringrdquo StatisticalMethodologyvol 5 no 6 pp 502ndash514 2008
[10] A A Ismail ldquoBayes estimation of Gompertz distributionparameters and acceleration factor under partially accelerated
10 Mathematical Problems in Engineering
life tests with type-I censoringrdquo Journal of Statistical Computa-tion and Simulation vol 80 no 11 pp 1253ndash1264 2010
[11] A A Ismail ldquoEstimating the parameters ofWeibull distributionand the acceleration factor from hybrid partially accelerated lifetestrdquo Applied Mathematical Modelling vol 36 no 7 pp 2920ndash2925 2012
[12] T A Abushal and A A Soliman ldquoEstimating the Paretoparameters under progressive censoring data for constant-partially accelerated life testsrdquo Journal of Statistical Computationand Simulation vol 85 no 5 pp 917ndash934 2015
[13] A A Ismail and A M Sarhan ldquoOptimal design of step-stresslife test with progressively type-II censored exponential datardquoInternational Mathematical Forum vol 4 no 37ndash40 pp 1963ndash1976 2009
[14] H Jeffreys The Theory of Probability Oxford University PressLondon UK 1998
[15] Q Guan Y Tang and A Xu ldquoObjective Bayesian analysis forbivariate Marshall-Olkin exponential distributionrdquo Computa-tional Statistics and Data Analysis vol 64 pp 299ndash313 2013
[16] M Han ldquoE-Bayesian estimation of failure probability and itsapplicationrdquoMathematical and ComputerModelling vol 45 no9-10 pp 1272ndash1279 2007
[17] M Han ldquoE-Bayesian estimation and hierarchical Bayesian esti-mation of failure probabilityrdquo Communications in StatisticsmdashTheory and Methods vol 40 no 18 pp 3303ndash3314 2011
[18] N L Johnson S Kotz and N Balakrishnan ContinuousUnivariate Distributions vol 1 Wiley-Interscience New YorkNY USA 2nd edition 1994
[19] AA Soliman ldquoEstimations for Paretomodel using general pro-gressive censored data and asymmetric lossrdquo Communicationsin StatisticsTheory andMethods vol 37 no 8ndash10 pp 1353ndash13702008
[20] H T Davis and M L Feldstein ldquoThe generalized Pareto law asa model for progressively censored survival datardquo Biometrikavol 66 no 2 pp 299ndash306 1979
[21] C M Harris ldquoThe Pareto distribution as a queue servicedisciplinerdquoOperations Research vol 16 no 2 pp 307ndash313 1968
[22] M BourguignonH Saulo andRN Fernandez ldquoA newPareto-type distribution with applications in reliability and incomedatardquo Physica A Statistical Mechanics and Its Applications vol457 pp 166ndash175 2016
[23] M Doostparast M G Akbari and N Balakrishnan ldquoBayesiananalysis for the two-parameter Pareto distribution based onrecord values and timesrdquo Journal of Statistical Computation andSimulation vol 81 no 11 pp 1393ndash1403 2011
[24] J O Berger Statistical Decision Theory and Bayesian AnalysisSpringer New York NY USA 2nd edition 1985
[25] D N P Murthy M Xie and R Jiang ldquoWeibull modelsrdquo inWiley Series in Probability and Statistics John Wiley and SonsHoboken NJ USA 2004
[26] W B Nelson ldquoStatistical methods for accelerated life test datathe inverse power law modelrdquo General Electric Co Tech Rep71-C011 Schenectady New York NY USA 1970
[27] N Balakrishnan and R A Sandhu ldquoA simple simulationalalgorithm for generating progressive Type-II censored samplesrdquoThe American Statistician vol 49 no 2 pp 229ndash230 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
Table 1 The AREs and MSEs of all estimates for differentm n and for censoring Scheme 1
(119899119898) Para OBEARE (MSE)
BEARE (MSE)
E-BE1ARE (MSE)
E-BE2ARE (MSE)
(40 10) 120582 00526 (00073) 00570 (00081) 00519 (00052) 00498 (0050)1205791 00360 (00048) 00378 (00076) 00392 (00075) 00382 (00065)1205792 00474 (00115) 00485 (00138) 00512 (00128) 00508 (00107)
(40 15) 120582 00514 (00069) 00551 (00067) 00515 (00048) 00495 (00044)1205791 00353 (00042) 00364 (00065) 00387 (00062) 00374 (00061)1205792 00462 (00107) 00472 (00120) 00504 (00112) 00500 (00099)
(40 20) 120582 00489 (00057) 00548 (00059) 00505 (00044) 00495 (00041)1205791 00336 (00032) 00341 (00056) 00372 (00051) 00365 (00049)1205792 00458 (00104) 00463 (00113) 00500 (00110) 00491 (00095)
(45 10) 120582 00513 (00070) 00564 (00080) 00514 (00050) 00498 (00048)1205791 00357 (00046) 00375 (00074) 00390 (00071) 00383 (00063)1205792 00471 (00111) 00486 (00127) 00505 (00115) 00512 (00100)
(45 15) 120582 00510 (00064) 00549 (00058) 00514 (00047) 00491 (00046)1205791 00350 (00040) 00362 (00061) 00386 (00059) 00370 (00060)1205792 00460 (00101) 00470 (00115) 00501 (00110) 00500 (00098)
(45 20) 120582 00484 (00056) 00542 (00054) 00504 (00041) 00493 (00038)1205791 00334 (00030) 00338 (00051) 00370 (00048) 00362 (00048)1205792 00454 (00100) 00462 (00111) 00497 (00105) 00489 (00087)
(50 10) 120582 00511 (00068) 00563 (00076) 00511 (00047) 00494 (00046)1205791 00353 (00044) 00373 (00071) 00387 (00070) 00380 (00061)1205792 00468 (00109) 00483 (00118) 00502 (00112) 00510 (00094)
(50 15) 120582 00509 (00062) 00546 (00058) 00512 (00045) 00492 (00045)1205791 00348 (00037) 00359 (00060) 00382 (00058) 00368 (00058)1205792 00459 (00095) 00468 (00110) 00500 (00107) 00500 (00093)
(50 20) 120582 00481 (00054) 00541 (00051) 00503 (00040) 00491 (00035)1205791 00330 (00027) 00335 (00047) 00368 (00042) 00369 (00043)1205792 00445 (00096) 00462 (00106) 00497 (00102) 00488 (00072)
Step 4 For 1198731 + 1 le 119894 le 119898 generate 11990521119894 by solvingthe equation 119880119894 = 1 minus 12057912057211 1205911205721(120582minus1)119905minus120572112058221119894 and generate 11990522119894 bysolving119882119894 = 1 minus 12057912057222 1205911205722(120582minus1)119905minus120572212058222119894 Again take the minimumof (11990521119894 11990522119894) as 119905lowast119894 as well as the corresponding index of theminimum (if 11990521119894 lt 11990522119894 set 119888lowast119894 = 1 else set 119888lowast119894 = 2)Step 5 For1198731 + 1 le 119894 le 119898 let 119905119894 = 119905lowast119894 and 119888119894 = 119888lowast119894
Then (1199051 1198881 1198771) (119905119898 119888119898 119877119898) is the required sample
We assumed that the occurrence time of the jth (119895 = 1 2)failure cause of tested product follows Pareto distributions119875(120572 120579) = 119875(2113 0067) and 119875(120572 120579) = 119875(182 0051) underuse stress levels 1198780 and 120582 = 20 and 120591 = 011 the parametersof prior density are taken as 119886 = 08 119887 = 20 1205731 = 0851205732 = 06 1198891 = 18 and 1198892 = 21 Considering different119898 119899 and censoring schemes we generate samples accordingto the above steps We replicate the process of evaluating1000 times in each case and compute the average relativeerrors (AREs) and mean squared errors (MSEs) ARE andMSE of the estimation of the parameter 120579 can be calculatedaccording to the following formulas
ARE () = 110001000sum119896=1
1003816100381610038161003816100381610038161003816120579 minus (119896)1003816100381610038161003816100381610038161003816120579
MSE () = 110001000sum119896=1
(120579 minus (119896))2 (28)
where (119896) is the 119896th estimator for the parameter 120579We calculate the AREs and MSEs of the objective
Bayesian estimates (OBE) Bayesian estimates (BE) andE-Bayesian estimates (E-BE) respectively The simulationresults are presented in Tables 1ndash3 In Tables 1ndash3 we denotethe E-BE based on the first prior as E-BE1 and the E-BE basedon the second prior as E-BE2
FromTables 1ndash3 the following observations can bemade
(1) For fixed n as m increases the AREs and MSEsdecrease for all estimates
(2) In terms of the values of AREs and MSEs the OBEperformances are better than that of BE and E-BE2
8 Mathematical Problems in Engineering
Table 2 The AREs and MSEs of all estimates for differentm n and for censoring Scheme 2
(119899 119898) Para OBEARE (MSE)
BEARE (MSE)
E-BE1ARE (MSE)
E-BE2ARE (MSE)
(40 10) 120582 00531 (00080) 00576 (00084) 00521 (00055) 00501 (00053)1205791 00365 (00051) 00381 (00081) 00398 (00079) 00386 (00067)1205792 00478 (00118) 00488 (00142) 00511 (00125) 00516 (00117)
(40 15) 120582 00516 (00071) 00551 (00082) 00515 (00051) 00495 (00049)1205791 00357 (00046) 00364 (00080) 00387 (00064) 00374 (00061)1205792 00465 (00113) 00472 (00138) 00504 (00114) 00500 (00112)
(40 20) 120582 00489 (00077) 00548 (00078) 00505 (00058) 00495 (00056)1205791 00336 (00044) 00341 (00075) 00372 (00056) 00365 (00053)1205792 00458 (00110) 00463 (00133) 00500 (00112) 00491 (00110)
(45 10) 120582 00527 (00079) 00564 (00079) 00511 (00052) 00501 (00051)1205791 00365 (00048) 00380 (00077) 00394 (00075) 00382 (00065)1205792 00474 (00115) 00485 (00133) 00509 (00122) 00512 (00114)
(45 15) 120582 00513 (00067) 00549 (00075) 00513 (00050) 00492 (00048)1205791 00348 (00037) 00364 (00074) 00382 (00062) 00370 (00058)1205792 00464 (00112) 00470 (00131) 00501 (00108) 00487 (00107)
(45 20) 120582 00486 (00065) 00546 (00071) 00501 (00057) 00492 (00045)1205791 00332 (00034) 00341 (00070) 00372 (00056) 00365 (00051)1205792 00451 (00110) 00457 (00127) 00492 (00104) 00490 (00104)
(50 10) 120582 00521 (00075) 00560 (00076) 00508 (00046) 00500 (00043)1205791 00362 (00046) 00380 (00075) 00392 (00074) 00381 (00063)1205792 00471 (00113) 00483 (00131) 00504 (00120) 00510 (00112)
(50 15) 120582 00504 (00065) 00546 (00074) 00512 (00050) 00491 (00045)1205791 00354 (00036) 00364 (00072) 00382 (00061) 00370 (00055)1205792 00464 (00110) 00470 (00130) 00501 (00105) 00496 (00104)
(50 20) 120582 00475 (00059) 00536 (00072) 00491 (00048) 00487 (00044)1205791 00324 (00029) 00332 (00170) 00366 (00052) 00356 (00051)1205792 00443 (00107) 00450 (00127) 00483 (00102) 00481 (00100)
performances are better than that of E-BE1 under thedifferent progressive censoring schemes
(3) Under the different progressive censoring schemesfor all estimates the values of AREs are less than 006and the values of MSEs are less than 0015 Hence theresults of all estimates are satisfied
6 Conclusions
In this paper the Bayesian analysis for the step-stress partiallyaccelerated competing failuremodel fromPareto distributionis investigated under Type II progressive censoring Based onthe noninformative prior and informative prior distributionswe derive the objective Bayesian estimates and Bayesianestimates of the unknown parameters and acceleration factorrespectively Considering such case that the hyperparametersare unknown and follow certain distributions the E-Bayesianmethod is utilized to estimate the unknown parameters andacceleration factor We analyze two real data sets to illustratethe application of the Pareto distribution in engineering
practice Finally the comparisons of the efficiencies of theobjective Bayesian estimates Bayesian estimates and corre-sponding E-Bayesian estimates are conducted It is concludedthat the OBE performances are better than that of BE andE-BE2 performances are better than that of E-BE1 underthe different progressive censoring schemes In addition wediscuss the effects of sample sizes and different progressivecensoring schemes on the estimates The simulation resultsdemonstrate that for fixed n the AREs and MSEs of allestimates decrease with119898 increasing
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The work is supported by the National Natural ScienceFoundation of China (71401134 71571144 and 71171164)The Natural Science Basic Research Program of ShaanxiProvince (2015JM1003) and The International Cooperation
Mathematical Problems in Engineering 9
Table 3 The AREs and MSEs of all estimates for differentm n and for censoring Scheme 3
(119899 119898) Para OBEARE (MSE)
BEARE (MSE)
E-BE1ARE (MSE)
E-BE2ARE (MSE)
(40 10) 120582 00535 (00079) 00578 (00082) 00521 (00056) 00510 (00054)1205791 00358 (00053) 00385 (00083) 00399 (00082) 00390 (00072)1205792 00481 (00129) 00491 (00145) 00512 (00130) 00520 (00122)
(40 15) 120582 00523 (00072) 00551 (00080) 00520 (00052) 00501 (00051)1205791 00364 (00048) 00371 (00077) 00396 (00076) 00380 (00071)1205792 00471 (00122) 00480 (00142) 00511 (00129) 00512 (00120)
(40 20) 120582 00491 (00066) 00552 (00078) 00513 (00051) 00504 (00050)1205791 00342 (00043) 00350 (00075) 00381 (00071) 00373 (00068)1205792 00465 (00120) 00469 (00140) 00507 (00124) 00499 (00118)
(45 10) 120582 00532 (00075) 00571 (00079) 00521 (00052) 00504 (00049)1205791 00363 (00051) 00381 (00081) 00398 (00081) 00390 (00067)1205792 00480 (00127) 00491 (00141) 00512 (00128) 00520 (00120)
(45 15) 120582 00518 (00068) 00551 (00075) 00520 (00050) 00501 (00046)1205791 00358 (00050) 00371 (00075) 00391 (00076) 00381 (00064)1205792 00467 (00118) 00478 (00137) 00510 (00123) 00510 (00117)
(45 20) 120582 00491 (00060) 00552 (00072) 00513 (00047) 00450 (00042)1205791 00340 (00048) 00345 (00069) 00381 (00065) 00371 (00062)1205792 00462 (00115) 00470 (00129) 00503 (00120) 00493 (00113)
(50 10) 120582 00530 (00074) 00571 (00067) 00520 (00050) 00502 (00045)1205791 00353 (00050) 00373 (00078) 00387 (00079) 00380 (00064)1205792 00468 (00124) 00483 (00140) 00502 (00124) 00510 (00119)
(50 15) 120582 00510 (00067) 00546 (00064) 00512 (00056) 00492 (00051)1205791 00348 (00042) 00359 (00075) 00382 (00076) 00368 (00062)1205792 00459 (00120) 00468 (00137) 00500 (00121) 00500 (00115)
(50 20) 120582 00481 (00065) 00541 (00062) 00503 (00053) 00491 (00050)1205791 00330 (00040) 00335 (00072) 00368 (00073) 00369 (00058)1205792 00454 (00116) 00462 (00134) 00497 (00118) 00485 (00113)
and Exchanges in Science and Technology Program ofShaanxi Province (2016KW-033)
References
[1] D Han and D Kundu ldquoInference for a step-stress model withcompeting risks for failure from the generalized exponentialdistribution under type-I censoringrdquo IEEE Transactions onReliability vol 64 no 1 pp 31ndash43 2015
[2] S Roy and C Mukhopadhyay ldquoMaximum likelihood analysisof multi-stress accelerated life test data of series systems withcompeting log-normal causes of failurerdquo Journal of Risk andReliability vol 229 no 2 pp 119ndash130 2015
[3] X P Zhang J Z Shang X Chen C H Zhang and Y S WangldquoStatistical inference of accelerated life testing with dependentcompeting failures based on copula theoryrdquo IEEE Transactionson Reliability vol 63 no 3 pp 764ndash780 2014
[4] Y-M Shi L Jin C Wei and H-B Yue ldquoConstant-stressaccelerated life test with competing risks under progressivetype-II hybrid censoringrdquo Advanced Materials Research vol712ndash715 pp 2080ndash2083 2013
[5] A Xu and Y Tang ldquoObjective Bayesian analysis of acceleratedcompeting failure models under Type-I censoringrdquo Computa-tional Statistics andData Analysis vol 55 no 10 pp 2830ndash28392011
[6] MWu Y Shi and Y Sun ldquoInference for accelerated competingfailure models fromWeibull distribution under Type-I progres-sive hybrid censoringrdquo Journal of Computational and AppliedMathematics vol 263 pp 423ndash431 2014
[7] N Balakrishnan andD Han ldquoExact inference for a simple step-stress model with competing risks for failure from exponentialdistribution under type-II censoringrdquo Journal of StatisticalPlanning and Inference vol 138 no 12 pp 4172ndash4186 2008
[8] D Han and N Balakrishnan ldquoInference for a simple step-stressmodel with competing risks for failure from the exponentialdistribution under time constraintrdquo Computational Statisticsand Data Analysis vol 54 no 9 pp 2066ndash2081 2010
[9] A M Abd-Elfattah A S Hassan and S G Nassr ldquoEstimationin step-stress partially accelerated life tests for the Burr typeXII distribution using type I censoringrdquo StatisticalMethodologyvol 5 no 6 pp 502ndash514 2008
[10] A A Ismail ldquoBayes estimation of Gompertz distributionparameters and acceleration factor under partially accelerated
10 Mathematical Problems in Engineering
life tests with type-I censoringrdquo Journal of Statistical Computa-tion and Simulation vol 80 no 11 pp 1253ndash1264 2010
[11] A A Ismail ldquoEstimating the parameters ofWeibull distributionand the acceleration factor from hybrid partially accelerated lifetestrdquo Applied Mathematical Modelling vol 36 no 7 pp 2920ndash2925 2012
[12] T A Abushal and A A Soliman ldquoEstimating the Paretoparameters under progressive censoring data for constant-partially accelerated life testsrdquo Journal of Statistical Computationand Simulation vol 85 no 5 pp 917ndash934 2015
[13] A A Ismail and A M Sarhan ldquoOptimal design of step-stresslife test with progressively type-II censored exponential datardquoInternational Mathematical Forum vol 4 no 37ndash40 pp 1963ndash1976 2009
[14] H Jeffreys The Theory of Probability Oxford University PressLondon UK 1998
[15] Q Guan Y Tang and A Xu ldquoObjective Bayesian analysis forbivariate Marshall-Olkin exponential distributionrdquo Computa-tional Statistics and Data Analysis vol 64 pp 299ndash313 2013
[16] M Han ldquoE-Bayesian estimation of failure probability and itsapplicationrdquoMathematical and ComputerModelling vol 45 no9-10 pp 1272ndash1279 2007
[17] M Han ldquoE-Bayesian estimation and hierarchical Bayesian esti-mation of failure probabilityrdquo Communications in StatisticsmdashTheory and Methods vol 40 no 18 pp 3303ndash3314 2011
[18] N L Johnson S Kotz and N Balakrishnan ContinuousUnivariate Distributions vol 1 Wiley-Interscience New YorkNY USA 2nd edition 1994
[19] AA Soliman ldquoEstimations for Paretomodel using general pro-gressive censored data and asymmetric lossrdquo Communicationsin StatisticsTheory andMethods vol 37 no 8ndash10 pp 1353ndash13702008
[20] H T Davis and M L Feldstein ldquoThe generalized Pareto law asa model for progressively censored survival datardquo Biometrikavol 66 no 2 pp 299ndash306 1979
[21] C M Harris ldquoThe Pareto distribution as a queue servicedisciplinerdquoOperations Research vol 16 no 2 pp 307ndash313 1968
[22] M BourguignonH Saulo andRN Fernandez ldquoA newPareto-type distribution with applications in reliability and incomedatardquo Physica A Statistical Mechanics and Its Applications vol457 pp 166ndash175 2016
[23] M Doostparast M G Akbari and N Balakrishnan ldquoBayesiananalysis for the two-parameter Pareto distribution based onrecord values and timesrdquo Journal of Statistical Computation andSimulation vol 81 no 11 pp 1393ndash1403 2011
[24] J O Berger Statistical Decision Theory and Bayesian AnalysisSpringer New York NY USA 2nd edition 1985
[25] D N P Murthy M Xie and R Jiang ldquoWeibull modelsrdquo inWiley Series in Probability and Statistics John Wiley and SonsHoboken NJ USA 2004
[26] W B Nelson ldquoStatistical methods for accelerated life test datathe inverse power law modelrdquo General Electric Co Tech Rep71-C011 Schenectady New York NY USA 1970
[27] N Balakrishnan and R A Sandhu ldquoA simple simulationalalgorithm for generating progressive Type-II censored samplesrdquoThe American Statistician vol 49 no 2 pp 229ndash230 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
Table 2 The AREs and MSEs of all estimates for differentm n and for censoring Scheme 2
(119899 119898) Para OBEARE (MSE)
BEARE (MSE)
E-BE1ARE (MSE)
E-BE2ARE (MSE)
(40 10) 120582 00531 (00080) 00576 (00084) 00521 (00055) 00501 (00053)1205791 00365 (00051) 00381 (00081) 00398 (00079) 00386 (00067)1205792 00478 (00118) 00488 (00142) 00511 (00125) 00516 (00117)
(40 15) 120582 00516 (00071) 00551 (00082) 00515 (00051) 00495 (00049)1205791 00357 (00046) 00364 (00080) 00387 (00064) 00374 (00061)1205792 00465 (00113) 00472 (00138) 00504 (00114) 00500 (00112)
(40 20) 120582 00489 (00077) 00548 (00078) 00505 (00058) 00495 (00056)1205791 00336 (00044) 00341 (00075) 00372 (00056) 00365 (00053)1205792 00458 (00110) 00463 (00133) 00500 (00112) 00491 (00110)
(45 10) 120582 00527 (00079) 00564 (00079) 00511 (00052) 00501 (00051)1205791 00365 (00048) 00380 (00077) 00394 (00075) 00382 (00065)1205792 00474 (00115) 00485 (00133) 00509 (00122) 00512 (00114)
(45 15) 120582 00513 (00067) 00549 (00075) 00513 (00050) 00492 (00048)1205791 00348 (00037) 00364 (00074) 00382 (00062) 00370 (00058)1205792 00464 (00112) 00470 (00131) 00501 (00108) 00487 (00107)
(45 20) 120582 00486 (00065) 00546 (00071) 00501 (00057) 00492 (00045)1205791 00332 (00034) 00341 (00070) 00372 (00056) 00365 (00051)1205792 00451 (00110) 00457 (00127) 00492 (00104) 00490 (00104)
(50 10) 120582 00521 (00075) 00560 (00076) 00508 (00046) 00500 (00043)1205791 00362 (00046) 00380 (00075) 00392 (00074) 00381 (00063)1205792 00471 (00113) 00483 (00131) 00504 (00120) 00510 (00112)
(50 15) 120582 00504 (00065) 00546 (00074) 00512 (00050) 00491 (00045)1205791 00354 (00036) 00364 (00072) 00382 (00061) 00370 (00055)1205792 00464 (00110) 00470 (00130) 00501 (00105) 00496 (00104)
(50 20) 120582 00475 (00059) 00536 (00072) 00491 (00048) 00487 (00044)1205791 00324 (00029) 00332 (00170) 00366 (00052) 00356 (00051)1205792 00443 (00107) 00450 (00127) 00483 (00102) 00481 (00100)
performances are better than that of E-BE1 under thedifferent progressive censoring schemes
(3) Under the different progressive censoring schemesfor all estimates the values of AREs are less than 006and the values of MSEs are less than 0015 Hence theresults of all estimates are satisfied
6 Conclusions
In this paper the Bayesian analysis for the step-stress partiallyaccelerated competing failuremodel fromPareto distributionis investigated under Type II progressive censoring Based onthe noninformative prior and informative prior distributionswe derive the objective Bayesian estimates and Bayesianestimates of the unknown parameters and acceleration factorrespectively Considering such case that the hyperparametersare unknown and follow certain distributions the E-Bayesianmethod is utilized to estimate the unknown parameters andacceleration factor We analyze two real data sets to illustratethe application of the Pareto distribution in engineering
practice Finally the comparisons of the efficiencies of theobjective Bayesian estimates Bayesian estimates and corre-sponding E-Bayesian estimates are conducted It is concludedthat the OBE performances are better than that of BE andE-BE2 performances are better than that of E-BE1 underthe different progressive censoring schemes In addition wediscuss the effects of sample sizes and different progressivecensoring schemes on the estimates The simulation resultsdemonstrate that for fixed n the AREs and MSEs of allestimates decrease with119898 increasing
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The work is supported by the National Natural ScienceFoundation of China (71401134 71571144 and 71171164)The Natural Science Basic Research Program of ShaanxiProvince (2015JM1003) and The International Cooperation
Mathematical Problems in Engineering 9
Table 3 The AREs and MSEs of all estimates for differentm n and for censoring Scheme 3
(119899 119898) Para OBEARE (MSE)
BEARE (MSE)
E-BE1ARE (MSE)
E-BE2ARE (MSE)
(40 10) 120582 00535 (00079) 00578 (00082) 00521 (00056) 00510 (00054)1205791 00358 (00053) 00385 (00083) 00399 (00082) 00390 (00072)1205792 00481 (00129) 00491 (00145) 00512 (00130) 00520 (00122)
(40 15) 120582 00523 (00072) 00551 (00080) 00520 (00052) 00501 (00051)1205791 00364 (00048) 00371 (00077) 00396 (00076) 00380 (00071)1205792 00471 (00122) 00480 (00142) 00511 (00129) 00512 (00120)
(40 20) 120582 00491 (00066) 00552 (00078) 00513 (00051) 00504 (00050)1205791 00342 (00043) 00350 (00075) 00381 (00071) 00373 (00068)1205792 00465 (00120) 00469 (00140) 00507 (00124) 00499 (00118)
(45 10) 120582 00532 (00075) 00571 (00079) 00521 (00052) 00504 (00049)1205791 00363 (00051) 00381 (00081) 00398 (00081) 00390 (00067)1205792 00480 (00127) 00491 (00141) 00512 (00128) 00520 (00120)
(45 15) 120582 00518 (00068) 00551 (00075) 00520 (00050) 00501 (00046)1205791 00358 (00050) 00371 (00075) 00391 (00076) 00381 (00064)1205792 00467 (00118) 00478 (00137) 00510 (00123) 00510 (00117)
(45 20) 120582 00491 (00060) 00552 (00072) 00513 (00047) 00450 (00042)1205791 00340 (00048) 00345 (00069) 00381 (00065) 00371 (00062)1205792 00462 (00115) 00470 (00129) 00503 (00120) 00493 (00113)
(50 10) 120582 00530 (00074) 00571 (00067) 00520 (00050) 00502 (00045)1205791 00353 (00050) 00373 (00078) 00387 (00079) 00380 (00064)1205792 00468 (00124) 00483 (00140) 00502 (00124) 00510 (00119)
(50 15) 120582 00510 (00067) 00546 (00064) 00512 (00056) 00492 (00051)1205791 00348 (00042) 00359 (00075) 00382 (00076) 00368 (00062)1205792 00459 (00120) 00468 (00137) 00500 (00121) 00500 (00115)
(50 20) 120582 00481 (00065) 00541 (00062) 00503 (00053) 00491 (00050)1205791 00330 (00040) 00335 (00072) 00368 (00073) 00369 (00058)1205792 00454 (00116) 00462 (00134) 00497 (00118) 00485 (00113)
and Exchanges in Science and Technology Program ofShaanxi Province (2016KW-033)
References
[1] D Han and D Kundu ldquoInference for a step-stress model withcompeting risks for failure from the generalized exponentialdistribution under type-I censoringrdquo IEEE Transactions onReliability vol 64 no 1 pp 31ndash43 2015
[2] S Roy and C Mukhopadhyay ldquoMaximum likelihood analysisof multi-stress accelerated life test data of series systems withcompeting log-normal causes of failurerdquo Journal of Risk andReliability vol 229 no 2 pp 119ndash130 2015
[3] X P Zhang J Z Shang X Chen C H Zhang and Y S WangldquoStatistical inference of accelerated life testing with dependentcompeting failures based on copula theoryrdquo IEEE Transactionson Reliability vol 63 no 3 pp 764ndash780 2014
[4] Y-M Shi L Jin C Wei and H-B Yue ldquoConstant-stressaccelerated life test with competing risks under progressivetype-II hybrid censoringrdquo Advanced Materials Research vol712ndash715 pp 2080ndash2083 2013
[5] A Xu and Y Tang ldquoObjective Bayesian analysis of acceleratedcompeting failure models under Type-I censoringrdquo Computa-tional Statistics andData Analysis vol 55 no 10 pp 2830ndash28392011
[6] MWu Y Shi and Y Sun ldquoInference for accelerated competingfailure models fromWeibull distribution under Type-I progres-sive hybrid censoringrdquo Journal of Computational and AppliedMathematics vol 263 pp 423ndash431 2014
[7] N Balakrishnan andD Han ldquoExact inference for a simple step-stress model with competing risks for failure from exponentialdistribution under type-II censoringrdquo Journal of StatisticalPlanning and Inference vol 138 no 12 pp 4172ndash4186 2008
[8] D Han and N Balakrishnan ldquoInference for a simple step-stressmodel with competing risks for failure from the exponentialdistribution under time constraintrdquo Computational Statisticsand Data Analysis vol 54 no 9 pp 2066ndash2081 2010
[9] A M Abd-Elfattah A S Hassan and S G Nassr ldquoEstimationin step-stress partially accelerated life tests for the Burr typeXII distribution using type I censoringrdquo StatisticalMethodologyvol 5 no 6 pp 502ndash514 2008
[10] A A Ismail ldquoBayes estimation of Gompertz distributionparameters and acceleration factor under partially accelerated
10 Mathematical Problems in Engineering
life tests with type-I censoringrdquo Journal of Statistical Computa-tion and Simulation vol 80 no 11 pp 1253ndash1264 2010
[11] A A Ismail ldquoEstimating the parameters ofWeibull distributionand the acceleration factor from hybrid partially accelerated lifetestrdquo Applied Mathematical Modelling vol 36 no 7 pp 2920ndash2925 2012
[12] T A Abushal and A A Soliman ldquoEstimating the Paretoparameters under progressive censoring data for constant-partially accelerated life testsrdquo Journal of Statistical Computationand Simulation vol 85 no 5 pp 917ndash934 2015
[13] A A Ismail and A M Sarhan ldquoOptimal design of step-stresslife test with progressively type-II censored exponential datardquoInternational Mathematical Forum vol 4 no 37ndash40 pp 1963ndash1976 2009
[14] H Jeffreys The Theory of Probability Oxford University PressLondon UK 1998
[15] Q Guan Y Tang and A Xu ldquoObjective Bayesian analysis forbivariate Marshall-Olkin exponential distributionrdquo Computa-tional Statistics and Data Analysis vol 64 pp 299ndash313 2013
[16] M Han ldquoE-Bayesian estimation of failure probability and itsapplicationrdquoMathematical and ComputerModelling vol 45 no9-10 pp 1272ndash1279 2007
[17] M Han ldquoE-Bayesian estimation and hierarchical Bayesian esti-mation of failure probabilityrdquo Communications in StatisticsmdashTheory and Methods vol 40 no 18 pp 3303ndash3314 2011
[18] N L Johnson S Kotz and N Balakrishnan ContinuousUnivariate Distributions vol 1 Wiley-Interscience New YorkNY USA 2nd edition 1994
[19] AA Soliman ldquoEstimations for Paretomodel using general pro-gressive censored data and asymmetric lossrdquo Communicationsin StatisticsTheory andMethods vol 37 no 8ndash10 pp 1353ndash13702008
[20] H T Davis and M L Feldstein ldquoThe generalized Pareto law asa model for progressively censored survival datardquo Biometrikavol 66 no 2 pp 299ndash306 1979
[21] C M Harris ldquoThe Pareto distribution as a queue servicedisciplinerdquoOperations Research vol 16 no 2 pp 307ndash313 1968
[22] M BourguignonH Saulo andRN Fernandez ldquoA newPareto-type distribution with applications in reliability and incomedatardquo Physica A Statistical Mechanics and Its Applications vol457 pp 166ndash175 2016
[23] M Doostparast M G Akbari and N Balakrishnan ldquoBayesiananalysis for the two-parameter Pareto distribution based onrecord values and timesrdquo Journal of Statistical Computation andSimulation vol 81 no 11 pp 1393ndash1403 2011
[24] J O Berger Statistical Decision Theory and Bayesian AnalysisSpringer New York NY USA 2nd edition 1985
[25] D N P Murthy M Xie and R Jiang ldquoWeibull modelsrdquo inWiley Series in Probability and Statistics John Wiley and SonsHoboken NJ USA 2004
[26] W B Nelson ldquoStatistical methods for accelerated life test datathe inverse power law modelrdquo General Electric Co Tech Rep71-C011 Schenectady New York NY USA 1970
[27] N Balakrishnan and R A Sandhu ldquoA simple simulationalalgorithm for generating progressive Type-II censored samplesrdquoThe American Statistician vol 49 no 2 pp 229ndash230 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
Table 3 The AREs and MSEs of all estimates for differentm n and for censoring Scheme 3
(119899 119898) Para OBEARE (MSE)
BEARE (MSE)
E-BE1ARE (MSE)
E-BE2ARE (MSE)
(40 10) 120582 00535 (00079) 00578 (00082) 00521 (00056) 00510 (00054)1205791 00358 (00053) 00385 (00083) 00399 (00082) 00390 (00072)1205792 00481 (00129) 00491 (00145) 00512 (00130) 00520 (00122)
(40 15) 120582 00523 (00072) 00551 (00080) 00520 (00052) 00501 (00051)1205791 00364 (00048) 00371 (00077) 00396 (00076) 00380 (00071)1205792 00471 (00122) 00480 (00142) 00511 (00129) 00512 (00120)
(40 20) 120582 00491 (00066) 00552 (00078) 00513 (00051) 00504 (00050)1205791 00342 (00043) 00350 (00075) 00381 (00071) 00373 (00068)1205792 00465 (00120) 00469 (00140) 00507 (00124) 00499 (00118)
(45 10) 120582 00532 (00075) 00571 (00079) 00521 (00052) 00504 (00049)1205791 00363 (00051) 00381 (00081) 00398 (00081) 00390 (00067)1205792 00480 (00127) 00491 (00141) 00512 (00128) 00520 (00120)
(45 15) 120582 00518 (00068) 00551 (00075) 00520 (00050) 00501 (00046)1205791 00358 (00050) 00371 (00075) 00391 (00076) 00381 (00064)1205792 00467 (00118) 00478 (00137) 00510 (00123) 00510 (00117)
(45 20) 120582 00491 (00060) 00552 (00072) 00513 (00047) 00450 (00042)1205791 00340 (00048) 00345 (00069) 00381 (00065) 00371 (00062)1205792 00462 (00115) 00470 (00129) 00503 (00120) 00493 (00113)
(50 10) 120582 00530 (00074) 00571 (00067) 00520 (00050) 00502 (00045)1205791 00353 (00050) 00373 (00078) 00387 (00079) 00380 (00064)1205792 00468 (00124) 00483 (00140) 00502 (00124) 00510 (00119)
(50 15) 120582 00510 (00067) 00546 (00064) 00512 (00056) 00492 (00051)1205791 00348 (00042) 00359 (00075) 00382 (00076) 00368 (00062)1205792 00459 (00120) 00468 (00137) 00500 (00121) 00500 (00115)
(50 20) 120582 00481 (00065) 00541 (00062) 00503 (00053) 00491 (00050)1205791 00330 (00040) 00335 (00072) 00368 (00073) 00369 (00058)1205792 00454 (00116) 00462 (00134) 00497 (00118) 00485 (00113)
and Exchanges in Science and Technology Program ofShaanxi Province (2016KW-033)
References
[1] D Han and D Kundu ldquoInference for a step-stress model withcompeting risks for failure from the generalized exponentialdistribution under type-I censoringrdquo IEEE Transactions onReliability vol 64 no 1 pp 31ndash43 2015
[2] S Roy and C Mukhopadhyay ldquoMaximum likelihood analysisof multi-stress accelerated life test data of series systems withcompeting log-normal causes of failurerdquo Journal of Risk andReliability vol 229 no 2 pp 119ndash130 2015
[3] X P Zhang J Z Shang X Chen C H Zhang and Y S WangldquoStatistical inference of accelerated life testing with dependentcompeting failures based on copula theoryrdquo IEEE Transactionson Reliability vol 63 no 3 pp 764ndash780 2014
[4] Y-M Shi L Jin C Wei and H-B Yue ldquoConstant-stressaccelerated life test with competing risks under progressivetype-II hybrid censoringrdquo Advanced Materials Research vol712ndash715 pp 2080ndash2083 2013
[5] A Xu and Y Tang ldquoObjective Bayesian analysis of acceleratedcompeting failure models under Type-I censoringrdquo Computa-tional Statistics andData Analysis vol 55 no 10 pp 2830ndash28392011
[6] MWu Y Shi and Y Sun ldquoInference for accelerated competingfailure models fromWeibull distribution under Type-I progres-sive hybrid censoringrdquo Journal of Computational and AppliedMathematics vol 263 pp 423ndash431 2014
[7] N Balakrishnan andD Han ldquoExact inference for a simple step-stress model with competing risks for failure from exponentialdistribution under type-II censoringrdquo Journal of StatisticalPlanning and Inference vol 138 no 12 pp 4172ndash4186 2008
[8] D Han and N Balakrishnan ldquoInference for a simple step-stressmodel with competing risks for failure from the exponentialdistribution under time constraintrdquo Computational Statisticsand Data Analysis vol 54 no 9 pp 2066ndash2081 2010
[9] A M Abd-Elfattah A S Hassan and S G Nassr ldquoEstimationin step-stress partially accelerated life tests for the Burr typeXII distribution using type I censoringrdquo StatisticalMethodologyvol 5 no 6 pp 502ndash514 2008
[10] A A Ismail ldquoBayes estimation of Gompertz distributionparameters and acceleration factor under partially accelerated
10 Mathematical Problems in Engineering
life tests with type-I censoringrdquo Journal of Statistical Computa-tion and Simulation vol 80 no 11 pp 1253ndash1264 2010
[11] A A Ismail ldquoEstimating the parameters ofWeibull distributionand the acceleration factor from hybrid partially accelerated lifetestrdquo Applied Mathematical Modelling vol 36 no 7 pp 2920ndash2925 2012
[12] T A Abushal and A A Soliman ldquoEstimating the Paretoparameters under progressive censoring data for constant-partially accelerated life testsrdquo Journal of Statistical Computationand Simulation vol 85 no 5 pp 917ndash934 2015
[13] A A Ismail and A M Sarhan ldquoOptimal design of step-stresslife test with progressively type-II censored exponential datardquoInternational Mathematical Forum vol 4 no 37ndash40 pp 1963ndash1976 2009
[14] H Jeffreys The Theory of Probability Oxford University PressLondon UK 1998
[15] Q Guan Y Tang and A Xu ldquoObjective Bayesian analysis forbivariate Marshall-Olkin exponential distributionrdquo Computa-tional Statistics and Data Analysis vol 64 pp 299ndash313 2013
[16] M Han ldquoE-Bayesian estimation of failure probability and itsapplicationrdquoMathematical and ComputerModelling vol 45 no9-10 pp 1272ndash1279 2007
[17] M Han ldquoE-Bayesian estimation and hierarchical Bayesian esti-mation of failure probabilityrdquo Communications in StatisticsmdashTheory and Methods vol 40 no 18 pp 3303ndash3314 2011
[18] N L Johnson S Kotz and N Balakrishnan ContinuousUnivariate Distributions vol 1 Wiley-Interscience New YorkNY USA 2nd edition 1994
[19] AA Soliman ldquoEstimations for Paretomodel using general pro-gressive censored data and asymmetric lossrdquo Communicationsin StatisticsTheory andMethods vol 37 no 8ndash10 pp 1353ndash13702008
[20] H T Davis and M L Feldstein ldquoThe generalized Pareto law asa model for progressively censored survival datardquo Biometrikavol 66 no 2 pp 299ndash306 1979
[21] C M Harris ldquoThe Pareto distribution as a queue servicedisciplinerdquoOperations Research vol 16 no 2 pp 307ndash313 1968
[22] M BourguignonH Saulo andRN Fernandez ldquoA newPareto-type distribution with applications in reliability and incomedatardquo Physica A Statistical Mechanics and Its Applications vol457 pp 166ndash175 2016
[23] M Doostparast M G Akbari and N Balakrishnan ldquoBayesiananalysis for the two-parameter Pareto distribution based onrecord values and timesrdquo Journal of Statistical Computation andSimulation vol 81 no 11 pp 1393ndash1403 2011
[24] J O Berger Statistical Decision Theory and Bayesian AnalysisSpringer New York NY USA 2nd edition 1985
[25] D N P Murthy M Xie and R Jiang ldquoWeibull modelsrdquo inWiley Series in Probability and Statistics John Wiley and SonsHoboken NJ USA 2004
[26] W B Nelson ldquoStatistical methods for accelerated life test datathe inverse power law modelrdquo General Electric Co Tech Rep71-C011 Schenectady New York NY USA 1970
[27] N Balakrishnan and R A Sandhu ldquoA simple simulationalalgorithm for generating progressive Type-II censored samplesrdquoThe American Statistician vol 49 no 2 pp 229ndash230 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
life tests with type-I censoringrdquo Journal of Statistical Computa-tion and Simulation vol 80 no 11 pp 1253ndash1264 2010
[11] A A Ismail ldquoEstimating the parameters ofWeibull distributionand the acceleration factor from hybrid partially accelerated lifetestrdquo Applied Mathematical Modelling vol 36 no 7 pp 2920ndash2925 2012
[12] T A Abushal and A A Soliman ldquoEstimating the Paretoparameters under progressive censoring data for constant-partially accelerated life testsrdquo Journal of Statistical Computationand Simulation vol 85 no 5 pp 917ndash934 2015
[13] A A Ismail and A M Sarhan ldquoOptimal design of step-stresslife test with progressively type-II censored exponential datardquoInternational Mathematical Forum vol 4 no 37ndash40 pp 1963ndash1976 2009
[14] H Jeffreys The Theory of Probability Oxford University PressLondon UK 1998
[15] Q Guan Y Tang and A Xu ldquoObjective Bayesian analysis forbivariate Marshall-Olkin exponential distributionrdquo Computa-tional Statistics and Data Analysis vol 64 pp 299ndash313 2013
[16] M Han ldquoE-Bayesian estimation of failure probability and itsapplicationrdquoMathematical and ComputerModelling vol 45 no9-10 pp 1272ndash1279 2007
[17] M Han ldquoE-Bayesian estimation and hierarchical Bayesian esti-mation of failure probabilityrdquo Communications in StatisticsmdashTheory and Methods vol 40 no 18 pp 3303ndash3314 2011
[18] N L Johnson S Kotz and N Balakrishnan ContinuousUnivariate Distributions vol 1 Wiley-Interscience New YorkNY USA 2nd edition 1994
[19] AA Soliman ldquoEstimations for Paretomodel using general pro-gressive censored data and asymmetric lossrdquo Communicationsin StatisticsTheory andMethods vol 37 no 8ndash10 pp 1353ndash13702008
[20] H T Davis and M L Feldstein ldquoThe generalized Pareto law asa model for progressively censored survival datardquo Biometrikavol 66 no 2 pp 299ndash306 1979
[21] C M Harris ldquoThe Pareto distribution as a queue servicedisciplinerdquoOperations Research vol 16 no 2 pp 307ndash313 1968
[22] M BourguignonH Saulo andRN Fernandez ldquoA newPareto-type distribution with applications in reliability and incomedatardquo Physica A Statistical Mechanics and Its Applications vol457 pp 166ndash175 2016
[23] M Doostparast M G Akbari and N Balakrishnan ldquoBayesiananalysis for the two-parameter Pareto distribution based onrecord values and timesrdquo Journal of Statistical Computation andSimulation vol 81 no 11 pp 1393ndash1403 2011
[24] J O Berger Statistical Decision Theory and Bayesian AnalysisSpringer New York NY USA 2nd edition 1985
[25] D N P Murthy M Xie and R Jiang ldquoWeibull modelsrdquo inWiley Series in Probability and Statistics John Wiley and SonsHoboken NJ USA 2004
[26] W B Nelson ldquoStatistical methods for accelerated life test datathe inverse power law modelrdquo General Electric Co Tech Rep71-C011 Schenectady New York NY USA 1970
[27] N Balakrishnan and R A Sandhu ldquoA simple simulationalalgorithm for generating progressive Type-II censored samplesrdquoThe American Statistician vol 49 no 2 pp 229ndash230 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
