Journal of ISOSS 2016 Vol. 2(2), 151-168 THE KUMARASWAMY … · 152 The Kumaraswamy...

© 2016 Journal of ISOSS 151

Journal of ISOSS

2016 Vol. 2(2), 151-168

THE KUMARASWAMY MARSHALL-OLKIN FRÉCHET DISTRIBUTION

WITH APPLICATIONS

Ahmed Z. Afify1, Haitham M. Yousof

1, Gauss M. Cordeiro

2,

Zohdy M. Nofal1 and Munir Ahmad

3

1 Department of Statistics, Mathematics and Insurance, Benha

University, Egypt. Email: [email protected]

[email protected]

[email protected] 2

Departamento de Estatística, Universidade Federal

de Pernambuco, Recife, Brazil. Email: [email protected]

3 National College of Business Administration and Economics,

Lahore, Pakistan. Email: [email protected]

ABSTRACT

The Fréchet distribution is an absolutely continuous distribution with wide applicability

in extreme value theory. Generalizing distributions is always precious for applied

statisticians and recent literature has suggested several ways of extending well-known

distributions. We propose a new lifetime model called the Kumaraswamy Marshall–Olkin

Fréchet distribution, which generalizes the Marshall–Olkin Fréchet distribution and at least

seventeen known and unknown lifetime models. Various properties of the new model are

explored including closed-forms expressions for moments, quantiles, generating function,

order statisics and Rényi entropy. The maximum likelihood method is used to estimate the

model parameters. We compare the flexibility of the proposed model with other related

distributions by means of two real data sets.

KEYWORDS

Generating Function, Marshall–Olkin Fréchet, Maximum Likelihood, Moment, Order

Statistics, Rényi Entropy.

1. INTRODUCTION

The procedure of expanding a family of distributions to add flexibility or to construct

covariate models is a well-known technique in the literature. In many applied sciences

like medicine, engineering and finance, among others, modeling and analyzing lifetime

data are crucial. Several lifetime distributions have been used to model these types of

data.

Marshall and Olkin (2007) defined and studied a new method of adding a new

parameter to an existing distribution. The generated distribution includes the baseline

distribution as a special case and gives more flexibility to model real data. For further and

comprehensive information about the Marshall–Olkin family of distributions, see

Barreto-Souza et al. (2013).

mailto:[email protected]

The Kumaraswamy Marshall–Olkin Fréchet Distribution with Applications 152

The Fréchet (Fr) distribution is an important distribution developed within the general

extreme value theory, which deals with the stochastic behaviour of the maximum and the

minimum of independent and identically distributed random variables. It has applications

ranging from accelerated life testing through to earthquakes, floods, horse racing, rainfall,

queues in supermarkets, sea waves and wind speeds. It has been applied to data on

characteristics of sea waves, wind speeds, etc. For more information about the Fr

distribution and its applications, see Kotz and Nadarajah (2000).

Given its applicability in extreme value theory, several extensions of the Fr distribution

were proposed in the literature. Some of its applications in various fields are given in

Harlow (2002). He showed that it is an important distribution for modeling statistical

behavior of material properties for a variety of engineering applications. Nadarajah and

Kotz (2008) discussed the sociological models based on Fr random variables. Zaharim et al.

(2009) applied the Fr distribution for analyzing the wind speed data. Mubarak (2011)

studied the Fr progressive type-II censored data with binomial removals.

Krishna et al. (2013) proposed some applications of the Marshall–Olkin Fr (MOFr)

distribution such as the reliability test plan for accepting or rejecting a lot where the

lifetime of the product follows the MOFr distribution and showed that it has also an

application in time series modeling.

Many authors defined generalizations of the Fr distribution. For example, Nadarajah

and Kotz (2003) pioneered the exponentiated Fr, Nadarajah and Gupta (2004) studied the

beta Fr, Mahmoud and Mandouh (2013) defined and studied the transmuted Fr, Krishna

et al. (2013) proposed the MOFr, Mead and Abd-Eltawab (2014) investigated the

Kumaraswamy Fr, Afify et al. (2015) defined the transmuted Marshall–Olkin Fr, Mead et

al. (2016) introduced the beta exponential Fr and Afify et al. (2016) proposed the Weibull

Fr distributions.

The cumulative distribution function (cdf) of the MOFr distribution (for ) is

given by

( ) [ .

/

] { ( ) [ .

/

]}

(1)

where is a scale parameter and and are shape parameters. The

corresponding probability density function (pdf) is given by

( ) [ .

/

] { ( ) [ .

/

]}

(2)

Our aim is to define and study a new lifetime model called the Kumaraswamy

Marshall-Olkin Fréchet (Kw-MOFr) distribution. The main feature of this model is that

two additional shape parameters are inserted in (1) to give more flexibility in the form of

the generated distribution. Following the Kumaraswamy-generalized (Kw-G) class

pioneered by Cordeiro and de Castro (2011), we construct the five-parameter Kw-MOFr

model. We give a comprehensive description of some mathematical properties of the new

distribution with the hope that it will attract wider applications in reliability, engineering

and other research.

A.Z. Afify et al. 153

For an arbitrary baseline cdf, ( ), Cordeiro and de Castro (2011) defined the Kw-G

family of distributions by the cdf and pdf given by

( ) , ( ) - (3)

and

( ) ( ) ( ) , ( ) - (4)

respectively, where ( ) ( ) and and are two additional positive shape

parameters. They can provide greater control over the weights in both tails and in the

center of the generated distribution and govern skewness. Clearly, for we

obtain the baseline distribution. If is a random variable with pdf (4), we write Kw-

G( ).

To this end, we start from the MOFr model to define the new Kw-MOFr model by

inserting (1) and (2) in equations (3) and (4). Then, we obtain the cdf and pdf of the Kw-

MOFr distribution.

The rest of the paper is organized as follows. In Section 2, we define the Kw-MOFr

distribution, provide some plots of its pdf and hazard rate function (hrf) and list some

special cases. We determine linear representations for the pdf and cdf in Section 3. Some

mathematical properties of the Kw-MOFr model including quantiles, ordinay moments

and cumulants, moment generating function (mgf), incomplete moments, mean

deviations, Rényi and q-entropies, moments of the residual life and reversed residual life

are obtained in Section 4. In Section 5, the order statistics and their moments are

determined. The maximum likelihood estimates (MLEs) of the unknown parameters are

explored in Section 6. We fit the new distribution to two real data sets to examine the

performance of the new model using nested and non-nested distributions in Section 7.

Finally, in Section 8, we provide some conclusions.

2. THE Kw-MOFr MODEL

The cdf of the Kw-MOFr distribution (for ) is given by

( ) { [ .

/

] { ( ) [ .

/

]}

}

(5)

and its pdf can be reduced to

( ) [ .

/

] { ( ) [ .

/

]}

{ [ .

/

] { ( ) [ .

/

]}

}

(6)

Henceforth, we denote a random variable having the pdf (6) by

Kw-MOFr( ).

The survival function (sf), hrf and cumulative hazard rate function (chrf) of are,

respectively, given by


( ) { [ .

/

] { ( ) [ .

/

]}

}

( ) [ .

/ ]{ ( ) [ .

/ ]}

[ .

/ ]{ ( ) [ .

/ ]}

and

( ) { [ .

/

] { ( ) [ .

/

]}

}

Plots of the Kw-MOFr pdf for some parameter values are displayed in Figure 1.

Figure 2 displays some possible shapes of the hrf of the Kw-MOFr model for selected

parameter values.

The Kw-MOFr model is a very flexible distribution that approaches different

distributions when its parameters are changed. The new distribution includes as special

cases seventeen well-known and other new models, namely the Kumaraswamy Fr (Kw-

Fr), Kumaraswamy inverse exponential (Kw-IE), Kumaraswamy inverse Rayleigh (Kw-

IR), Kumaraswamy Marshall–Olkin inverse exponential (Kw-MOIE), Kumaraswamy

Marshall–Olkin inverse Rayleigh (Kw-MOIR), MOFr, Marshall–Olkin inverse

exponential (MOIE), Marshall–Olkin inverse Rayleigh (MOIR), exponentiated Fr (EFr),

exponentiated inverse exponential (EIE), exponentiated inverse Rayleigh (EIR),

generalized Fr (GFr), generalized inverse exponential (GIE), generalized inverse

Rayleigh (GIR), Fr, inverse exponential (IE), inverse Rayleigh (IR) models. Table 1 lists

all these sub-models.

Figure 1: Plots of the Kw-MOFr Density Function


Table 1

Sub-Models of the Kw-MOFr Distribution

No. Distribution Author

1 Kw-Fr Mead and Abd-Eltawab (2014)

2 Kw-IE –

3 Kw-IR –

4 Kw-MOIE New

5 Kw-MOIR New

6 MOFr Krishna et al. (2013)

7 MOIE –

8 MOIR –

9 EFr Nadarajah and Kotz (2003)

10 EIE –

11 EIR –

12 GFr New

13 GIE New

14 GIR New

15 Fr Fréchet (1924)

16 IE Keller and Kamath (1982)

17 IR Trayer (1964)

Figure 2: Plots of the Kw-MOFr hrf


3. LINEAR REPRESENTATION

In this section, we provide linear representations for the pdf and cdf of the new

distribution. Consider a power series given by

( ) ∑

( )

( ) | | (7)

Applying expansion (7) to equation (6) gives

( ) ∑

( ) ( )

( ) [ ( ) .

/

]

{ ( ) [ .

/

]} ( )

(8)

Applying (7) again to (8), we obtain

( ) ∑

( ) ( ) ( )

( ) ( ) ( ) .

/

[ ( ) .

/

]

The last equation can be expressed as

( ) ∑ ( ) (9)

where ( ) is the Fr density with scale parameter ( ) and shape

parameter and is a constant term given by

( ) ( ) ( )

( ) ( ) ( ).

/

Therefore, the Kw-MOFr density is given by a linear combination of the Fr densities.

So, several of its structural properties can be derived from those of the Fr distribution.

Similarly, the cdf of in (5) admits a linear representation given by

( ) ∑ ( )

where ( ) is the cdf of the Fr model with scale parameter ( ) and

shape parameter .

These linear representations will be used to obtain some structural properties of the

new distribution.

4. SOME STATISTICAL PROPERTIES

Established algebraic expansions to determine some structural properties of the Kw-

MOFr distribution can be more efficient than computing those directly by numerical

integration of its density function. The statistical properties of the Kw-MOFr distribution

including moments, quantile function (qf), mgf, incomplete moments, mean deviations

and Rényi entropy are discussed in the following sections.


4.1 Moments

Let be a random variable having the Fr distribution with parameters and The

th moment of (for ) is given by .

/ and the th incomplete

moment of is given by ( ) (

.

/

), where ( ) ∫

is

the complete gamma function and ( ) ∫

is the lower incomplete

gamma function.

The th ordinary moment, say of is given by

( ) ∑

∫

( )

Thus, we obtain (for )

( ) .

/∑

( ) (10)

By setting in equation (10), we obtain the mean of The skewness and

kurtosis measures can be determined from the ordinary moments of using well-known

relationships.

The th central moment of , say , is just given by

( ) ∑

. / (

)

Then,

∑

. / (

) .

/ ∑

( )

The cumulants ( ) of follow from (10) as ∑

.

/ ,

respectively, where . Thus,

,

,

, etc. The skewness

and kurtosis

can be calculated from the third and fourth standardized cumulants.

Finally, the th descending factorial moment of is ( ) ( ( )) , ( )

( )- ∑ ( )

, where ( ) ( ) , ( ) - is the

Stirling number of the first kind.

4.2 Quantile and Generating Functions

The qf of is determined by inverting (5) as

( ) { ( √ √

√ √

)}

Simulating the Kw-MOFr random variable is straightforward. If is a uniform

variate in the unit interval ( ) the random variable ( ) follows (6).


The effects of the additional shape parameters , and on the skewness and

kurtosis of the new distribution can be based on quantile measures. The well-known

Bowley's skewness and Moors' kurtosis are, respectively, defined by

( ) ( ) ( )

( ) ( )

and

( ) ( ) ( ) ( )

( ) ( )

These measures are less sensitive to outliers and they exist even for distributions

without moments. Figure 3 displays some plots for the measures and of . They

indicate the variability of these measures on the shape parameters. It is clear from Figure

3 that the Kw-MOFr model is a right-skewed distribution.

Next, we provide the mgf of the Fr distribution as defined by Afify et al. (2016). We

can write the mgf of as

( ) ∫

( ) ( )

By expanding the first exponential and determining the integral, we obtain

( ) ∑

.

/

Figure 3: Skewness (left panel) and Kurtosis (right panel)

Plots of the Kw-MOFr Distribution

Consider the Wright generalized hypergeometric function defined by

[( ) ( )

( ) ( ) ] ∑

∏ ( )

∏

( )

Then, we can write ( ) as

( ) 0( )

1

Combining the last expression and (9), the mgf of reduces to


( ) ∑ 0

( )

( ) 1

4.3 Incomplete Moments

The th incomplete moment of , denoted by ( ), is given by ( )

∫

( ) . We can write from equation (9)

( ) ∑ ∫

( )

and using the lower incomplete gamma function, we obtain (for )

( ) ∑ ( ) (

( ) .

/

) (11)

The first incomplete moment of is immediately obtained from (11) by setting

. The main application of ( ) is related to the mean residual life and the mean

waiting time (also known as mean inactivity time) given by ( ) , ( )- ( ) and ( ) ( ) ( ) respectively.

Further, the amount of scatter in a population is evidently measured to some extent by

the totality of deviations from the mean and median. The mean deviations about the mean

, (| |)- and about the median , (| |)- of are given by

∫

|

| ( ) (

) ( ) and ∫

| | ( )

( ) respectively, where

( ) comes from (10) and ( ) is determined

by (5).

Another application of ( ) refers to the Lorenz and Bonferroni curves defined by

( ) ( ) and ( ) ( ) (

), respectively, where ( ) can be

computed for a given probability by inverting (5) numerically.

4.4 Rényi and -Entropies

Entropy refers to the amount of uncertainty associated with a random variable. The

Rényi entropy has numerous applications in information theory (for example,

classification, distribution identification problems, and statistical inference), computer

science (for example, average case analysis for random databases, pattern recognition,

and image matching) and econometrics, see Källberg et al. (2014). The Rényi entropy of

a random variable is defined by

( )

(∫

( ) )

Then, the Rényi entropy for the Kw-MOFr model reduces to

( )

{ ∑

∫

( ) [ ( ) .

/

] ⏟

}

where

( ) and

( ) ( ) ( ) ( )

( ( ) ) ( ).

/


But

( )( ), ( )- ( )

where ( ) .

Finally, we obtain

( )

{∑

( ) ( )}

The -entropy, say ( ), is defined by

( )

* ,( ) ( )-+

4.5 Moments of the Residual and Reversed Residual Life

The th moment of the residual life, ( ) (( ) | ), ,

uniquely determines ( ) (see Navarro et al., 1998). It is given by ( )

( )∫

( ) ( )

Using equation (9), we can write

( )

( )∑

∫

( ) ( )

By using the binomial expansion and the upper incomplete gamma function defined

by ( ) ∫

, we obtain

( )

( )∑

( )

( ) ( ) ∑

(

( ) .

/

)

Another interesting function is the mean residual life (MRL) function at age defined

by ( ) ,( )| -, which represents the expected additional life length for a

unit which is alive at age . The MRL of follows by setting in the last equation.

The th moment of the reversed residual life, ( ) ,( ) | -, for

, uniquely determines ( ). The th moment of the residual life is

given by ( )

( )∫

( ) ( ).

Using equation (9), we can write

( )

( )∑

∫

( ) ( )

By using the binomial expansion and the lower incomplete gamma function, we

obtain

( )

( )∑

( )

( ) ( )

∑ (

( ) .

/

)

The mean inactivity time (MIT), also called mean reversed residual life (MRRL)

function, is defined by ( ) ,( )| -. It represents the waiting time elapsed


since the failure of an itemon condition that this failure had occurred in ( ). The MIT

of the Kw-MOFr distribution can be determined by setting in the last equation.

5. ORDER STATISTICS AND MOMENTS

Let be a random sample of size from the Kw-MOFr model with cdf and

pdf as given in (5) and (6), respectively. Let be the corresponding order

statistics. Then, the pdf of , denoted by ( ), is given by

( )

( ) { [ .

/

] }

( )

[ .

/

] { ( [ .

/

] )

}

where ,( ) ( ) -

The joint pdf of and , for , is given (for ) by

( ) ( ) ( ) ( ) , ( ) ( )- , ( )-

Then, by inserting (5) and (6) in the last equation, we obtain

( ) ( ) ( ) ( )( )

{, ( ) -[ ( )

]}

2, ( ) - [ ( )

] 3

* , ( ) - + *, ( )

- +

where

( ) ( ) ( ) , { ( ) [ .

/

]} and .

/

for

.

The pdf of can also be expressed by

( ) ( )

( )∑

( ) .

/ ( ) (12)

Therefore, we can write

( ) ∑ ( ) .

/

{ [ .

/

] ( ( ) [ .

/

])

}

By inserting (5) and the last equation in (12), we obtain

( ) ∑ ( ) ( ) (13)

where ( ) ( )is the Fr pdf with parameters and , ( ) - and


( )∑

∑

( ) ( )

( )

.

/ .

/

.

/ . ( )

/

So, the density function of the Kw-MOFr order statistics is a linear combination of Fr

densities. Based on equation (13), we can obtain some structural properties of from

those Fr properties.

The th moment of is given by

(

) ∑ ( ( )

) (14)

Based upon the moments (14), we can derive explicit expressions for the L-moments

of as infinite weighted linear combinations of suitable Fr expected values given by

∑

( ) .

/ ( )

6. ESTIMATION

Several approaches for parameter estimation were proposed in the statistical literature

but the maximum likelihood method is the most commonly employed. The MLEs enjoy

desirable properties for constructing confidence intervals. We investigate the estimation

of the parameters of the Kw-MOFr distribution by maximum likelihood for complete data

sets. Let ( ) be a random sample of this distribution with unknown

parameter vector ( ) Then, the log-likelihood function for , say

( ), is

( ) ( ) ( )∑ ∑

.

/

( )∑ ( ) ( )∑

, ( )

-

where ( )( ) and [ .

/

].

The MLE of can be determined by maximizing ( ) (for a given ) either

directly by using the Mathcad, R (optim function), SAS (PROC NLMIXED), Ox

program (sub-routine MaxBFGS) or by solving the nonlinear system obtained by

differentiating this equation and equating its five components to zero.

Let .

/ ( )

.

/ and ( ) .

/ .

/

.

The components of the score vector ( )

(

) are given by

( )∑

( )∑

( )( )

( )

.

/ ∑

( )∑ ( )( )

( )∑

[ ( ) ( )

]

( )

∑ ( )


∑

( ) ( )∑

( )

( )∑

( )

( )

, ( )

( )-

∑

.

/

∑ ( )∑

(

) ( )

( )

and

∑

, ( )

-

It is usually more convenient to use nonlinear optimization algorithms such as

the quasi-Newton algorithm to numerically maximize the log-likelihood function. For

interval estimation of the model parameters, we require the observed information

matrix ( ) * + (for ). Under standard regularity conditions, the

multivariate normal ( ( ) ) distribution can be used to construct approximate

confidence intervals for the model parameters. Here, ( ) is the total observed

information matrix evaluated at . Therefore, approximate ( ) confidence

intervals for and can be determined as:

√

√

√

√ and

√ , where

is the

upper th percentile of the standard normal distribution.

7. DATA ANALYSIS

In this section, we provide applications of the Kw-MOFr distribution using two real

data sets. We compare the proposed distribution with other related lifetime models,

namely: the MOFr, Kw-Fr, beta Fr (BFr), MOIE, MOIR, EFr and Fr distributions. The

pdf of the BFr model is given (for ) by

( )

( ) .

/

[ .

/

]

,

where and are positive parameters.

In order to compare the distributions, we consider some criteria like the maximized

log-likelihood ( ), Akaike information criterion ( ), consistent Akaike information

criterion ( ), Bayesian information criterion ( ) and Hannan-Quinn information

criterion ( ). The model with minimum values for or or or can

be chosen as the best model to fit the data.

The first data set from Bjerkedal (1960) consists of survival times for guinea pigs

injected with different doses of tubercle bacilli: 12, 15, 22, 24, 24, 32, 32, 33, 34, 38, 38,

43, 44, 48, 52, 53, 54, 54, 55, 56, 57, 58, 58, 59, 60, 60, 60, 60, 61, 62, 63, 65, 65, 67, 68,

70, 70, 72, 73, 75, 76, 76, 81, 83, 84, 85, 87, 91, 95, 96, 98, 99, 109, 110, 121, 127, 129,

131, 143, 146, 146, 175, 175, 211, 233, 258, 258, 263, 297, 341, 341, 376. These data

were previously studied by Krishna et al. (2013).


The values of , , , and for the fitted models are given in

Tables 2 and 4. The MLEs and their corresponding standard errors (in parentheses) of the

model parameters for the fitted Kw-MOIE, Kw-MOIR, Kw-Fr, MOFr, BFr, EFr and Fr

distributions to both data sets are listed in Tables 3 and 5. The figures in these tables are

obtained using the Mathcad program. The histogram and the estimated pdfs and cdfs for

the best fitted models to the survival time guinea pigs and strengths of cm glass fibres

are displayed in Figures 4 and 5, respectively.

Table 2

The and Statistics for the First Data Set

Model

Kw-MOFr

EFr

Kw-Fr

BFr

Kw-MOIE

Fr

MOFr

Kw-MOIR

Figure 4: Fitted pdfs and cdfs to the Survival Times for Guinea Pigs

The second data set obtained from Smith and Naylor (1987) consists of the strengths

of cm glass fibres, measured at the National Physical Laboratory, England.

Unfortunately, the units of measurement are not given in the paper. The observations

are: 0.55, 0.93,1.25, 1.36, 1.49, 1.52, 1.58, 1.61, 1.64, 1.68, 1.73, 1.81, 2 , 0.74, 1.04,

1.27, 1.39, 1.49, 1.53,1.59, 1. 61, 1.66, 1.68, 1.76, 1.82, 2.01, 0.77, 1.11, 1.28, 1.42, 1.5,

1.54, 1.6, 1.62, 1.66, 1.69,1.76, 1.84, 2.24, 0.81, 1.13, 1.29, 1.48, 1.5, 1.55, 1.61, 1.62,


1.66, 1.7, 1.77, 1.84, 0.84, 1.24, 1.3, 1.48, 1.51, 1.55, 1.61, 1.63, 1.67, 1.7, 1.78, 1.89.

These data were previously studied by Barreto-Souza et al. (2011) and Afify et al. (2015).

Table 3

MLEs and their Standard Errors for the First Data Set

Model Estimates

Kw-MOFr

( ) ( ) ( ) ( ) ( )

EFr – –

– ( ) ( ) – ( )

Kw-Fr –

– ( ) ( ) ( ) ( )

BFr –

– ( ) ( ) ( ) ( )

Kw-MOIE –

( ) – ( ) ( ) ( )

Fr – – –

– ( ) ( ) – –

MOFr – –

( ) ( ) ( ) – –

Kw-MOIR –

( ) – ( ) ( ) ( )

Table 4

The and Statistics for the Second Data Set

Model

Kw-MOFr

Kw-Fr

Kw-MOIE

EFr

BFr

Kw-MOIR

Fr

MOF


Based on the figures in Tables 2 and 4, we can compare the Kw-MOFr model with

the other fitted models. It is clear from these figures that the Kw-MOFr model gives the

lowest values for the , , and statistics (except for the Kw-Fr

model in the second data set) among all fitted models. The plots in Figures 3 and 4 also

indicate the same thing. So, the Kw-MOFr model could be chosen as the best model.

Figure 5: Fitted pdfs and cdfs to the Strengths of Glass Fibres

Table 5

MLEs and their Standard Errors for the Second Data Set

Model Estimates

Kw-MOFr

( ) ( ) ( ) ( ) ( )

Kw-Fr –

– ( ) ( ) ( ) ( )

Kw-MOIE –

( ) ( ) – ( ) ( )

EFr – –

– ( ) ( ) – ( )

BFr –

– ( ) ( ) ( ) ( )

Kw-MOIR –

( ) – ( ) ( ) ( )

Fr – – –

– ( ) ( ) – –

MOFr – –

( ) ( ) ( ) – –


8. CONCLUSIONS

We propose a new five-parameter lifetime model, called the Kumaraswamy

Marshall–Olkin Fréchet (Kw-MOFr) distribution, which extends the Marshall–Olkin

Fréchet (MOFr) (Krishna et al., 2013) distribution. We prove that the Kw-MOFr density

function can be given as a mixture of Fréchet densities. We obtain explicit expressions

for the ordinary and incomplete moments, quantile and generating functions, mean

deviations, moments of the residual life, moments of the reversed residual life and Rényi

and -entropies. We also investigate the density function of order statistics and their

moments. We discuss the maximum likelihood estimation of the model parameters. We

prove empirically that the Kw-MOFr model can give a better fit than other lifetime

models by means of two real data sets.

REFERENCES

1. Afify, A.Z., Hamedani, G.G., Ghosh, I. and Mead, M.E. (2015). The transmuted

Marshall-Olkin Fréchet distribution: properties and applications. International

Journal of Statistics and Probability, 4, 132-148.

2. Afify, A.Z., Yousof, H.M., Cordeiro, G.M., Ortega, E.M.M. and Nofal, Z.M. (2016).

The Weibull Fréchet distribution and its applications. Journal of Applied Statistics,

43, 2608-2626.

3. Barlow, R.E. and Proschan, F. (1996). Mathematical Theory of Reliability, Vol. 17 of

Classics in Applied Mathematics, Society for Industrial and Applied Mathematics

(SIAM), Philadelphia, PA.

4. Barreto-Souza, W., Cordeiro, G.M. and Simas, A.B. (2011). Some results for beta

Fréchet distribution. Communications in Statistics-Theory and Methods, 40, 798-811.

5. Barreto-Souza, W., Lemonte, A.J. and Cordeiro, G.M. (2013). General results for the

Marshall and Olkin's family of distributions. Annals of the Brazilian Academy of

Sciences, 85, 3-21.

6. Bjerkedal, T. (1960). Acquisition of resistance in guinea pigs infected with different

doses of virulent tubercle bacilli. Amer. J. Hyg., 72, 130-148.

7. Cordeiro, G.M. and de Castro, M. (2011). A new family of generalized distributions.

Journal of Statistical Computation and Simulation, 81, 883-898.

8. Fréchet, M. (1924). Sur la loi des erreurs dIobservation. Bulletin de la Société

Mathématique de Moscou, 33, 5-8.

9. Gupta, R.C. (1975). On characterization of distributions by conditional expectations.

Communications in Statistics-Theory and Methods, 4, 99-103.

10. Gupta, P.L. and Gupta, R.C. (1983). On the moments of residual life in reliability and

some characterization results. Communications in Statistics-Theory and Methods, 12,

449-461.

11. Harlow, D.G. (2002). Applications of the Fréchet distribution function. International

Journal of Materials and Product Technology, 17, 482-495

12. Källberga, D., Leonenkob, N. and Seleznjeva, O. (2014). Statistical estimation of

quadratic Rényi entropy for a stationary M-dependent sequence. Journal of

Nonparametric Statistics, 26, 385-411.

13. Keller, A.Z. and Kamath, A.R. (1982). Reliability analysis of CNC machine tools.

Reliability Engineering, 3, 449-473.


14. Kotz, S. and Nadarajah, S. (2000). Extreme value distributions: Theory and

applications. Imperial College Press, London.

15. Kotz, S. and Shanbhag, D.N. (1980). Some new approaches to probability

distributions. Adv. Appl. Prob., 12, 903-921.

16. Krishna, E., Jose, K.K., Alice, T. and Ristić, M.M. (2013). The Marshall-Olkin

Fréchet distribution. Communications in Statistics-Theory and Methods, 42,

4091-4107.

17. Krishna, E., Jose, K.K. and Ristić, M.M. (2013). Applications of Marshall–Olkin

Fréchet distribution. Communications in Statistics-Simulation and Computation,

42(1), 76-89.

18. Mahmoud, M.R. and Mandouh, R.M. (2013). On the transmuted Fréchet distribution.

Journal of Applied Sciences Research, 9, 5553-5561.

19. Marshall, A.W. and Olkin, I. (1997). A new method for adding a parameter to a

family of distributions with application to the exponential and Weibull families.

Biometrika, 84, 641-652.

20. Mead, M.E. and Abd-Eltawab A.R. (2014). A note on Kumaraswamy Fréchet

distribution. Aust. J. Basic & Appl. Sci., 8, 294-300.

21. Mead, M.E., Afify, A.Z., Hamedani, G.G. and Ghosh, I. (2016). The beta exponential

Fréchet distribution with applications. Austrian Journal of Statistics, forthcoming.

22. Mubarak, M. (2012). Parameter Estimation Based on the Frèchet Progressive Type II

Censored Data with Binomial Removals,” International Journal of Quality, Statistics,

and Reliability, 2012, Article ID 245910, 5 pages, 2012. doi:10.1155/2012/245910.

23. Nadarajah, S. and Gupta, A.K. (2004). The beta Fréchet distribution. Far East

Journal of Theoretical Statistics, 14, 15-24.

24. Nadarajah, S. and Kotz, S. (2003). The exponentiated exponential distribution,

Satistica, Available online at

http://interstat.statjournals.net/YEAR/2003/abstracts/0312001.php.

25. Nadarajah, S. and Kotz, S. (2008). Sociological models based on Fréchet random

variables. Quality and Quantity, 42, 89-95.

26. Navarro, J. Franco, M. and Ruiz, J.M. (1998). Characterization through moments of

the residual life and conditional spacing. The Indian Journal of Statistics, Series A,

60, 36-48.

27. Smith, R.L. and Naylor, J.C. (1987). A comparison of maximum likelihood and

Bayesian estimators for the three-parameter Weibull distribution. Appl. Statist., 36,

358-369.

28. Treyer, V.N. (1964). Doklady Acad, Nauk, Belorus, U.S.S.R.

29. Winkler, R.L., Roodman, G.M. and Britney, R.R. (1972). The determination of partial

moments. Management Science, 19, 290-296.

30. Zaharim, A., Najid, S.K., Razali, A.M. and Sopian, K. (2009). Analysing Malaysian

wind speed data using statistical distribution. In Proceedings of the 4th

IASME/WSEAS International conference on energy and environment, Cambridge,

UK.

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