Some new Families of Continuous Distributions Generated from...

Some new Families of Continuous

Distributions Generated from Burr XII Logit

By

Muhammad Arslan Nasir

(Roll No. 01 , Session 2013-16)

Registration No. 81/IU.PhD/2013

A thesis submitted to

The Islamia University of Bahawalpur

For the Partial Fulfilment of the degree of

Doctor of Philosophy in Statistics

January 2017

Department of Statistics

The Islamia University of Bahawalpur

BAHAWALPUR 63100, PAKISTAN www.iub.edu.pk

Declaration

I, Muhammad Arslan Nasir solemnly declare that the work done in this thesis entitled

” Some new Families of Continuous Distributions Generated from Burr XII Logit” is my

own and original otherwise acknowledged. This work has not been submitted as a whole

or in part for any other degree to any other university in Pakistan or abroad.

MUHAMMAD ARSLAN NASIR

Email: [email protected]

i

Plagiarism Undertaking

I Muhammad Arslan Nasir solemnly declare that research work presented in the thesis

titled ”Some new Families of Continuous Distributions Generated from Burr XII Logit”

is solely my research work with no significant contribution from any other person. Small

contribution/help wherever taken has been duly acknowledged and that complete thesis

has been written by me.

I understand the zero tolerance policy of the HEC and University ”The Islamia Uni-

versity of Bahawalpur” towards plagiarism. Therefore I as an Author of the above titled

thesis declare that no portion of my thesis has been plagiarized and any material used as

reference is properly referred/cited.

I undertake that if I am found guilty of any formal plagiarism in the above titled thesis

even after award of PhD degree, the University reserves the rights to withdraw/revoke

my PhD degree and that HEC and the University has the right to publish my name on the

HEC/University Website on which names of students are placed who submitted plagia-

rized thesis.

STUDENT /AUTHORSIGNATURE:

ii

Examination Committee

We confer the degree of Doctor of Philosophy (Ph.D.) in Statistics to Mr. Muhammad Ar-

slan Nasir on May 24, 2017.

DR. MUHAMMAD HUSSAIN TAHIR:

SUPERVISOR AND INTERNAL EXAMINER

PROFESSOR, DEPARTMENT OF STATISTICS, IUB

DR. MUHAMMAD AKRAM:

EXTERNAL EXAMINER

PROFESSOR(RTD.), DEPARTMENT OF STATISTICS, BZU, MULTAN.

DR. AHMED FAISAL SIDDIQI:

EXTERNAL EXAMINER

PROFESSOR, DEPARTMENT OF STATISTICS, UMT, LAHORE.

DR. SHAKIR ALI GHAZALI:

CHAIRMAN

PROFESSOR, DEPARTMENT OF STATISTICS, IUB

iii

Certificate from supervisor

It is to certify that Muhammad Arslan Nasir has completed this thesis/research work enti-

tled ”Some new Families of Continuous Distributions Generated from Burr XII Logit”

for the Doctor in Statistics under my supervision.

(Supervisor)

DR. M.H. TAHIR

Professor of Statistics, The Islamia University of Bahawalpur, Pakistan.

Email: [email protected]

iv

Abstract

This thesis is based on six chapters. In these chapters five new families of distributions

are introduced by using the Burr XII distribution. In Chapter 1, a brief introduction of

the existing families of distribution, the objectives and organization of this thesis are pre-

sented. In Chapter 2, Generalized Burr G family of distributions is proposed by using

the function of cdf − log[1 − G(x)]. In Chapter 3, Marshall-Olkin Burr G family of dis-

tributions is introduced by using odd Burr G family of distributions used as generator

proposed by Alizadeh et al. (2017). In chapter 4, odd Burr G Poisson family of distribution

is introduced by compounding odd Burr G family with zero truncated Poisson distribu-

tion. In Chapter 5, a new generalized Burr distribution based on the quantile function

following the method given by Aljarrah et al. (2014). In Chapter 6, Kumaraswamy odd

Burr G family of distributions is introduced using odd Burr G family as a generator. The

mathematical properties of these families are obtained, such as asymptotes and shapes,

infinite mixture representation of the densities of the families, rth moment, sth incomplete

moment, moment generating function, mean deviations, reliability and stochastic order-

ing, two entropies, Renyi and Shannon entropies. The explicit expression of distribution

ith order statistic is also obtained in terms of linear combination of baseline densities and

probability weighted moments. Model parameters are estimated by using the maximum

likelihood (ML) method for complete and censored samples. Special models are given for

each family, their plots of density and hazard rate functions are displayed. One special

model for each family is investigated in detail. Simulation studies are also carried out to

assess the validity of ML estimates of the model discussed in detail. Application on real

life data is done to check the performance of the proposed families.

v

Acknowledgments

First and foremost, I would like to thank Allah for giving me the strength and the will to

succeed.

I would like to thank my supervisor Dr. M. H. Tahir for his innovative guidance, dedi-

cation, knowledge, tremendous patience, constructive suggestions and enormous support

throughout this research and the writing in this thesis. His insights and words of encour-

agement has often inspired me and renewed my hopes for completing my Ph.D. research.

I am very much thankful to the Chairman Department of Statistics and other teachers at

the Department of Statistics, The Islamia University of Bahawalpur, Pakistan.

Many thanks to Farrukh Jamal (research fellow) for the support, unconditional friend-

ship, advice and for putting up with me all the time.

I would like to thank my Mother and Father, special thank to my brother Muhammad

Salman Atir and younger brothers Muhammad Hassan Yasir and Muhammad Fayzan

Shakir for their encouragement towards PhD studies. I would like to thank my beloved

wife for her moral sport towards PhD studies.

vi

Dedication

To my parents

vii

Contents

Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Undertaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Examination Committee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Certificate from supervisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

1 Introduction 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Well-established generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Some Extensions of Burr XII distribution . . . . . . . . . . . . . . . . . . . . . 4

1.3.1 Beta-Burr XII distribution . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.2 Kumaraswamy-Burr XII distribution . . . . . . . . . . . . . . . . . . . 5

1.3.3 McDonald-Burr XII distribution . . . . . . . . . . . . . . . . . . . . . 6

1.3.4 Marshall-Olkin-Burr XII distribution . . . . . . . . . . . . . . . . . . . 7

1.4 Objectives of the research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4.1 Plan of research work for thesis . . . . . . . . . . . . . . . . . . . . . . 8

2 Generalized Burr Family of Distributions 9

viii

Section 0.0 Chapter 0

2.1 Mathematical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.1 Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.2 Infinite mixture representation . . . . . . . . . . . . . . . . . . . . . . 10

2.1.3 Moments and moment generating function . . . . . . . . . . . . . . . 12

2.1.4 Reliability parameter and Stochastic ordering . . . . . . . . . . . . . 13

2.2 Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Estimation of parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.1 Estimation of parameters in case of complete samples . . . . . . . . . 16

2.3.2 Estimation of parameters in case of censored samples . . . . . . . . . 16

2.4 Special sub-models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4.1 Generalized Burr Normal (GBN) distribution . . . . . . . . . . . . . . 18

2.4.2 Generalized Burr Lomax (GBLx) distribution . . . . . . . . . . . . . . 19

2.4.3 Generalized Burr Exponentiated Exponential (GBEE) distribution . . 20

2.4.4 Generalized Burr Uniform (GBU) distribution . . . . . . . . . . . . . 21

2.5 Mathematical properties of GBU distribution . . . . . . . . . . . . . . . . . . 22

2.5.1 Simulation and Application . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Marshall Olkin Burr G Family of Distributions 29

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Infinite mixture representation . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3 Asymptotics and Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4.1 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4.2 The Stress-Strength reliability parameters . . . . . . . . . . . . . . . . 35

3.4.3 Stochastic ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35


3.6 Estimation of parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37


ix


3.6.2 Estimation of parameters in case of censored complete samples . . . 37

3.7 Special models of MOBG family . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.7.1 Marshall-Olkin Burr XII Frechet (MOBFr) distribution . . . . . . . . 39

3.7.2 Marshall-Olkin Burr XII log-logistic (MOBLL) distribution . . . . . . 39

3.7.3 Marshall-Olkin Burr XII-Weibull (MOBW) distribution . . . . . . . . 41

3.7.4 Marshall-Olkin Burr XII Lomax (MOBLx) distribution . . . . . . . . 42

3.8 Mathematical properties of MOBLx distribution . . . . . . . . . . . . . . . . 43

3.8.1 Simulation study of MOBLx distribution . . . . . . . . . . . . . . . . 45

3.8.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.8.3 Data set 3: Carbon Fibres data . . . . . . . . . . . . . . . . . . . . . . 46

3.8.4 Data set 4: Remission Times data . . . . . . . . . . . . . . . . . . . . . 47

3.9 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Odd Burr-G Poisson Family of distributions 50

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 Special models of OBGP family . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2.1 Odd Burr-Weibull Poisson (OBWP) distribution . . . . . . . . . . . . 52

4.2.2 Odd Burr Lomax Poisson (OBLxP) distribution . . . . . . . . . . . . 53

4.2.3 Odd Burr gamma Poisson distribution (OBGaP) . . . . . . . . . . . . 54

4.2.4 Odd Burr beta Poisson (OBBP) distribution . . . . . . . . . . . . . . 55

4.3 Some mathematical properties of OBGP family . . . . . . . . . . . . . . . . . 56

4.3.1 Infinite mixture representation . . . . . . . . . . . . . . . . . . . . . . 56

4.3.2 Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3.3 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4 Entropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59



4.6 Maximum Likelihood method . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.7 Properties of OBLP distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.7.1 Simulations study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.8 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

x


4.8.1 Data set 5: Failure times mechanical components . . . . . . . . . . . 69

4.9 Conclusions and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5 A New Generalized Burr Distribution based on quantile function 72

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.1.1 T-Burr{Lomax} Family of distributions . . . . . . . . . . . . . . . . . 74

5.1.2 T-Burr{log-Logistic} Family of distributions . . . . . . . . . . . . . . 74

5.1.3 T-Burr{Weibull} Family of distributions . . . . . . . . . . . . . . . . 75

5.2 Some properties of the T-Burr{Y} family of distributions . . . . . . . . . . . 75

5.2.1 Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2.2 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2.3 Entropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2.4 Mean Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3 Special Sub-Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3.1 The Gamma-Burr{Log-logistic} (GaBLL) distribution. . . . . . . . . 80

5.3.2 The Dagum-Burr{Weibull} (DBW) distribution. . . . . . . . . . . . . 80

5.3.3 The Weibull-Burr{Lomax} (WBLx) distribution. . . . . . . . . . . . . 81

5.4 Simulation and application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.4.1 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.4.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.4.3 Complete data set 6: Diameter-Thickness . . . . . . . . . . . . . . . . 85

5.4.4 Censored data set 7: Remission-Times . . . . . . . . . . . . . . . . . . 86

5.5 Conclusions and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6 Kumaraswamy Odd Burr XII Family of distributions 89

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.2 Infinite mixture representation . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.3 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.3.1 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.4 Entropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.5 stochastic ordering, moments ofresidual and reversed residual life . . . . . . 95

xi



6.5.2 Moments of Residual and Reversed residual life . . . . . . . . . . . . 96


6.7 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98


6.7.2 Estimation of parameters in case of censored complete samples . . . 99

6.8 Special Sub Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.8.1 The Kumaraswamy odd Burr-Frechet (KOBFr) distribution. . . . . . 101

6.8.2 The Kumaraswamy odd Burr-Lomax (KOBLx) distribution. . . . . . 101

6.8.3 The Kumaraswamy odd Burr-Dagum distribution. . . . . . . . . . . 103

6.8.4 The Kumaraswamy odd Burr-Gompertz (KOBGo) distribution. . . . 104

6.8.5 The Kumaraswamy odd Burr-uniform (KWOBU) distribution. . . . 106

6.9 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.9.1 Data Set 8: Carbon Fibers . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.9.2 Data Set 9: Birnbaum-Saunders . . . . . . . . . . . . . . . . . . . . . . 109

6.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

xii

List of Tables

1.1 Different W [G(x)] functions for special models of the T-X family. . . . . . . 4

2.1 Mean and MSE for the of the MLEs of the parameters of the GB-U model. . 25

2.2 MLEs and their standard errors (in parentheses) for the data set 1. . . . . . . 26

2.3 MLEs and their standard errors (in parentheses) for data set 2. . . . . . . . . 27

3.1 Estimated AEs, biases and MSEs of the MLEs of parameters of MOBLx dis-

tribution based on 500 simulations of with n=50, 100 and 300. . . . . . . . . 46

3.2 The parameter estimates and A* and W* values for data set 3 . . . . . . . . . 47

3.3 The parameter estimates and A* and W* values for data set 4 . . . . . . . . . 49

4.1 Mean, bias and MSEs of the estimates of the parameters of OBLxP for c = 10,

k = 0.06, λ = 4 and α = 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2 Mean, bias and MSEs of the estimates of the parameters of OBLxP model

for c = 10, k = 0.5, λ = 4 and α = 9. . . . . . . . . . . . . . . . . . . . . . . . 68

4.3 Mean, bias and MSEs of the estimates of the parameters of OBLxP for c =

0.5, k = 0.06, λ = 4 and α = 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.4 Mean, bias and MSE (Mean Square Error) of the estimates of the parameters

of OBLxP with c = 10, k = 0.06, λ = 0.5 and α = 9. . . . . . . . . . . . . . . . 69

4.5 MLEs and their standard errors for data set 5. . . . . . . . . . . . . . . . . . 70

4.6 Model adequacy measures A∗ and W∗ for data set 5. . . . . . . . . . . . . . 70

5.1 qfs. for different distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

xiii


5.2 Estimated AEs, biases and MSEs of the MLEs of parameters of WBLx distri-

bution based on 1000 simulations for n=100, 200 and 500. . . . . . . . . . . . 84

5.3 MLEs and their standard errors (in parentheses) for Data set 6 . . . . . . . . 85

5.4 The Value, W*, A*, KS, P-Value values for data Set 6 . . . . . . . . . . . . . . 86

5.5 MLEs and their standard errors for Data set 7 . . . . . . . . . . . . . . . . . . 87

6.1 MLEs and their standard errors for data set 1. . . . . . . . . . . . . . . . . . . 109

6.2 MLEs and their standard errors for data set 2. . . . . . . . . . . . . . . . . . . 110

xiv

List of Figures

2.1 Plots of (a) density and (b) hrf for the GBN distribution with different pa-

rameter values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Plots of the (a) density and (b) hrf for the GBLx distribution with different

parameter values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Plots of the (a) density and (b) hrf for the GBEE distribution with different


2.4 Plots of the (a) density and (b) hrf for the GBU distribution with different


2.5 The estimated pdfs and cdfs of GBU and other competitive models for data

set 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6 The estimated pdfs and cdfs of GBU and other competitive models for data

set 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1 Plots of (a) density and (b) hrf for MOBFr distribution for different paramet-

ric values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2 Plots of (a) density and (b) hrf for MOBLL distribution for different para-

metric values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3 Plots of (a) density and (b) hrf for MOBW distribution for different paramet-

ric values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4 Plots of (a) density and (b) hrf for MOBLx distribution for different para-

metric values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.5 Plots of estimated pdf (a) and (c), cdf (b) and (d) for data set 3 and data set 4. 48

xv


4.1 Plots of (a) density and (b) hrf of OBWP distribution for some parameter

values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 Plots of (a) density and (b) hrf of OBLxP distribution for some parameter

values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3 Plots of (a) density and (b) hrf of OBGaP distribution for some parameter

values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.4 Plots of (a) density and (b) hrf of OBBP distribution for some parameter values 56

4.5 Plots of estimated pdf and cdf of OBLxP distribution . . . . . . . . . . . . . 70

5.1 Plots of (a) density and (b) hrf of GaBLL distribution . . . . . . . . . . . . . 80

5.2 Plots of (a) density and (b) hrf of DBW distribution . . . . . . . . . . . . . . 81

5.3 Plots of (a) density and (b) hrf of WBLx distribution . . . . . . . . . . . . . . 82

5.4 Estimated (a) pdfs and (b) cdfs for data set 6. . . . . . . . . . . . . . . . . . . 87

5.5 Plots of estimated cdf for censored data set 7. . . . . . . . . . . . . . . . . . . 88

6.1 Plots of (a) density and (b) hrf for KwOBuFr distribution for different pa-

rameter values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.2 Plots of (a) density and (b) hrf for KOBLx distribution with different para-

metric values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.3 Plots of (a) density and (b) hrf for KOBD distribution for different parameter

values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.4 Plots of (a) density and (b) hrf for KOBGo distribution for different param-

eter values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.5 Plots of pdf and hrf for KwOBuU distribution with different parametric val-

ues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.6 Plots of estimated pdf and cdf for data set 1. . . . . . . . . . . . . . . . . . . 110

6.7 Plots of estimated pdf and cdf for data set 2. . . . . . . . . . . . . . . . . . . 111

xvi

Chapter 1

Introduction

1.1 Introduction

In modern Statistics, the role of distribution theory is very influential. The statistical mod-

eling of the phenomenon, the applications or the validity of data is impossible without

choosing the proper mathematical form of the model (the probability distribution). In old

practice, proposing a new distribution or its generalization is solely based on suggesting

a different functional form through differential or mathematical equation. Various system

(or families) of distributions have been proposed in literature.

Burr (1942) introduced twelve different forms of cumulative distribution functions for

modeling lifetime data or survival data. Three members of the Burr family Viz. the Burr

types XII, III and X distributions are important because they are inherently more flexible

than the Weibull distribution. The Burr types XII, III and X distributions cover a much

larger area of the skewness kurtosis plane than the Weibull distribution (Rodriguez, 1977;

Tadikamalla, 1980). The Burr XII distribution has received increased attention in literature

due to application in physics, actuarial studies, reliability and applied statistics. The Burr

XII distribution also offers a wide range of functions of their parameters such as reliability,

hazard rate and mode under various conditions.

The Burr XII (BXII) distribution is a unimodal and has non-monotone hazard function.

The use of Burr XII as a lifetime model is appropriate and useful in applied statistics, espe-

cially in survival analysis and actuarial studies. Further Burr XII distribution contains the

1


shape characteristics of the normal, log-normal, gamma, logistic and exponential (Pearson

type X) distributions, as well as a significant portion of the Pearson types I (beta), II, III

(gamma), V, VII, IX and XII families. Other particular cases of the Burr XII include Fisher-

F, inverted beta, Lomax, Pareto and the log-logistic distributions. It is therefore observable

that the versatility and flexibility of the Burr XII distribution make it quite attractive as a

tentative and empirical model for data whose underlying distribution is unknown.

The cumulative distribution function (cdf) and probability density function (pdf) of the

two-parameter Burr XII distribution are, respectively, given by

FBXII(x) = 1− (1 + xc)−k (1.1)

and

fBXII(x) = ckxc−1 (1 + xc)−k−1 , (1.2)

where x > 0 and c > 0, k > 0 are parameters.

The cdf and pdf of the two-parameter Burr type III distribution are, respectively, given by

FBIII(x) =(1 + x−c

)−k (1.3)

and

fBIII(x) = ckx−c−1(1 + x−c

)−k−1, (1.4)

where x > 0 and c > 0, k > 0 are parameters.

The cdf and pdf of the two-parameter Burr type X distribution are, respectively, given by

FBX(x) = (1− e−(λx)2)θ (1.5)

and

fBX(x) = 2λ2 x θ e−(λx)2 (1− e−(λx)2)θ−1, (1.6)

where x > 0 and λ > 0, θ > 0 are parameters.

2


1.2 Well-established generators

Marshall and Olkin (1997) first suggested adding one parameter to the survival function

G(x) = 1 − G(x), where G(x) is the cumulative distribution function (cdf) of the baseline

distribution.

Gupta et al. (1998) added one parameter to the cdf, G(x), of the baseline distribution

to define the exponentiated-G (“exp-G” for short) class of distributions based on Lehmann-

type alternatives (see Lehmann, 1953).

Eugene et al. (2002) and Jones (2004) defined the beta-generated (beta-G) class from the

logit of the beta distribution. Further works on generalized distributions were Kumaraswamy-

G (Kw-G) by Cordeiro and de Castro (2011), McDonald-G (Mc-G) by Alexander et al.

(2012), gamma-G type 1 by Zografos and Balakrishanan (2009), gamma-G type 2 by Ristic

and Balakrishanan (2012), and Amini et al. (2014), odd-gamma-G type 3 by Torabi and

Montazari (2012), logistic-G by Torabi and Montazari (2014), odd exponentiated gener-

alized (odd exp-G) by Cordeiro et al. (2013), transformed-transformer (T-X) (Weibull-X

and gamma-X) by Alzaatreh et al. (2013), exponentiated T-X by Alzaghal et al. (2013),

odd Weibull-G by Bourguignon et al. (2014), exponentiated half-logistic by Cordeiro et al.

(2014a), logistic-X by Tahir et al. (2014a), new Weibull-G by Tahir et al. (2014b), T-X{Y}-

quantile based approach by Aljarrah et al. (2014) and T-R{Y} by Alzaatreh et al. (2014).

Let r(t) be the probability density function (pdf) of a random variable T ∈ [a, b] for

−∞ ≤ a < b < ∞ and let W [G(x)] be a function of the cumulative distribution function

(cdf) of a random variable X such that W [G(x)] satisfies the following conditions:

(i) W [G(x)] ∈ [a, b],

(ii) W [G(x)] is differentiable and monotonically non-decreasing, and

(iii) W [G(x)] → a as x → −∞ and W [G(x)] → b as x →∞.

(1.7)

Recently, Alzaatreh et al. (2013) defined the T-X family of distributions by

F (x) =∫ W [G(x)]

ar(t) dt, (1.8)

where W [G(x)] satisfies the condition (1.7). The pdf corresponding to Eq. (1.8) is given by

f(x) ={

d

dxW [G(x)]

}r {W [G(x)]} . (1.9)

3


In Table 1.1, we provide the W [G(x)] functions for some members of the T-X family of

distributions.

Table 1.1: Different W [G(x)] functions for special models of the T-X family.

S.No. W[G(x)] Range of T Members of T-X family

1 G(x) [0, 1] Beta-G (Eugene et al., 2002)

Kw-G type 1 (Cordeiro and de Castero, 2011)

Mc-G (Alexander et al., 2012)

Exp-G (Kw-G type 2) (Cordeiro et al., 2013)

3 - log [1−G(x)] (0,∞) Gamma-G Type-1 (Zografos and Balakrishnan, 2009)

Gamma-G Type-1 (Amini et al., 2014)

Weibull-X (Alzaatreh et al., 2013)

Gamma-X (Alzaatreh et al., 2013)

4 - log [1−Gα(x)] (0,∞) Exponentiated T-X (Alzaghal et al., 2013)

5 G(x)1−G(x) (0,∞) Gamma-G Type-3 (Torabi and Montazeri, 2012)

Weibull-G (Bourguinion et al., 2014)

6 log[ G(x)

1−G(x)

](−∞,∞) Logistic-G (Torabi and Montazeri, 2014)

7 log {- log [1−G(x)]} (−∞,∞) Logistic-X (Tahir et al., 2014a)

1.3 Some Extensions of Burr XII distribution

In litrature, four Extensions of Burr XII distributions are available.

4


1.3.1 Beta-Burr XII distribution

First the well-established generator beta-G is considered, which was introduced by Eugene

et al. (2002) and further discussed by Jones (2004).

For any arbitrary baseline pdf g(x) and cdf G(x), the cdf and pdf of beta-G class of

distributions are,respectively, given by

F (x) = IG(x)(a, b) (1.10)

and

f(x) =1

B(a, b)g(x)

{G(x)

}a−1 {1−G(x)

}b−1 (1.11)

where a > 0 and b > 0 and are both shape parameters. B(a, b) =1∫0

xa−1 (1− x)b−1 dx and

Bx(a, b) =x∫0

xa−1 (1− x)b−1 dx

Paranaiba et al. (2011) introduced a five parameter beta-Burr distribution by using beta-

G class defined in Eq. (1.10) and Eq. (1.11). The pdf and cdf are given, respectively, as

F (x) = I1−[1+(x/s)c]−k(a, b) (1.12)

and

f(x) =1

B(a, b)c k xc−1

sc

{1−

[1 +

(x

s

)c]−k}a−1 [

1 +(x

s

)c]−(kb+1). (1.13)

1.3.2 Kumaraswamy-Burr XII distribution

For a baseline random variable having pdf g(x) and cdf G(x), Cordeiro and de Castro

(2011) defined the two-parameter Kumaraswamy-G class. The cdf and pdf are defined by

F (x) = 1− [1−G(x)a]b (1.14)

and

f(x) = a b g(x) [G(x)]a−1 [1−G(x)a]b−1 , (1.15)

where g(x) = dG(x)/dx and a > 0 and b > 0 are two additional shape parameters whose

role are to govern skewness and tail weights.

5


Paranaiba et al. (2013) introduced a five parameter Kumaraswamy Burr (Kw-Burr) dis-

tribution by using Kw-G class defined in Eq. (1.14) and Eq. (1.15). The pdf and cdf of

Kw-Burr XII distribution are given, respectively, as

F (x) = 1−{

1−[1−

{1 +

(x

s

)c}−k]a}b

(1.16)

and

f(x) = a b c k s−c xc−1[1 +

(x

s

)c]−k−1{

1−[1 +

(x

s

)c]−k}a−1

×[1−

{1−

[1 +

(x

s

)c]−k}a]b−1

. (1.17)

1.3.3 McDonald-Burr XII distribution

For any arbitrary baseline pdf g(x) and cdf G(x), Alexander et al. (2012) defined the cdf

and pdf of McDonald-G (Mc-G) class of distributions as

F (x) = IG(x)c(ac−1, b) (1.18)

and

f(x) =c

B(ac−1, b)g(x) {G(x)}a−1 {1−G(x)c}b−1 . (1.19)

Gomes et al. (2013) proposed a six parameter McDonald-Burr XII(Mc-Burr XII) distri-

bution by using Mc-G class defined in Eq. (1.19) and Eq.(1.18). The pdf and cdf Mc-Burr

XII are respectively, given by

F (x) = I[1−[1+(x/s)α]−β]α(ac−1, b) (1.20)

and

f(x) =c

B(ac−1, b)α β

s

(x

s

)α−1 [1 +

(x

s

)α]−β−1{

1−[1 +

(x

s

)α]−β}α−1

×[1−

{1−

[1 +

(x

s

)α]−β}c]b−1

. (1.21)

6


1.3.4 Marshall-Olkin-Burr XII distribution

Marshall-Olkin (1997) proposed a flexible class of distribution. The cdf and pdf of Marshall-

Olkin extended(MOE) family are given by

F (x) =G(x)

1− (1− α)[1−G(x)](1.22)

and

f(x) =α g(x){

1− (1− α)[1−G(x)]}2 , (1.23)

where α > 0 is a shape (or tilt) parameter.

Al-Sariari et al. (2014) proposed a three parameter Marshall Olkin extended Burr XII

(MOEBXII) distribution. The cdf and the pdf of MOEBXII are given, respectively, as

F (x) =1− (1 + xc)−k

[1− (1− α) (1 + xc)−k

]2 (1.24)

and

f(x) =α c k xc−1 (1 + xc)−k−1

[1− (1− α) (1 + xc)−k

]2 , x > 0, (1.25)

where α, c, k > 0.

1.4 Objectives of the research

The main objectives of our research are:

• To propose new G-class based on Burr XII distribution.

• To check the flexibility of the proposed family of distributions, mathematically and

analytically.

• To investigate useful mathematical properties such as reliability properties, mean

residual and mean waiting time, quantile function, moments, incomplete moments,

probability weighted moments, moment generating function, entropies (Shannon,

Renyi), order statistics, parameter estimation etc.

7


• To investigate mathematical properties of some of the special model(s) of the pro-

posed class.

• To obtain stress strength reliability parameter and stochastic ordering etc.

• To report usefulness of the proposed G-classes distribution to the real data sets.

1.4.1 Plan of research work for thesis

The plan of this research work is to propose several G-class or generators from Burr XII

logit as follows:

• In Chapter 2, Generalized Burr-G class of distribution is proposed using− log {1−G(x)}generator and its properties are investigated.

• In Chapter 3, Marshall-Olkin Burr-G family of distributions is proposed and studied.

• In Chapter 4, Odd Burr-G Poisson family of distribution is introduced and studied.

• In Chapter 5, T-Burr{Y} family of distribution is proposed by using the quantile

function approach pioneered by Aljarrah et al. (2014).

• In Chapter 6, Kumaraswamy-odd Burr G family of distributions is proposed and its

mathematical properties are obtained.

8

Chapter 2

Generalized Burr Family of

Distributions

In this chapter, Generalized Burr-G (GB-G) family of distributions using the generator

− log {1−G(x)} is proposed, which is the quantile function (qf) of the standard exponen-

tial distribution. The cdf of the new GB-G family is given by

F (x; c, k, ξ) =

− log G(x;ξ)∫

0

r(t) dt. (2.1)

If r(t) = c k tc−1 (1 + tc)−k is the pdf of the BXII distribution, then

F (x; c, k, ξ) =

− log G(x;ξ)∫

0

c k tc−1 (1 + tc)−k dt = 1− (1 +{− log G(x; ξ)

}c)−k. (2.2)

The pdf corresponding to Eq. (2.2) is given by

f(x; c, k, ξ) = c kg(x; ξ)

1−G(x; ξ){− log G(x; ξ)

}c−1 (1 +{− log G(x; ξ)

}c)−k−1. (2.3)

Henceforth, a random variable with density (2.3) is denoted by X ∼ GBG(c, k, ξ).

The qf Q(u) can be determined by inverting Eq. (2.2) as

Qx(u) = G−1

(1− e−[(1−u)−

1k−1]

1c

), (2.4)

9


where QG(u) = G−1(u) is the baseline quantile function.

The failure rate (or hazard rate) is the frequency with which an engineered system or com-

ponent fails, expressed. The failure rate of a system usually depends on time. The hazard

rate function (hrf) of the GBG family given as

h(t; ξ) =c k g(t; ξ) {− log (1−G(t; ξ))}c−1

(1−G(t; ξ)) [1 + {− log (1−G(t; ξ))}c]

.

2.1 Mathematical Properties

Here, some mathematical properties of the GBG family are studied.

2.1.1 Shapes

The shapes of the density and hazard rate functions can be described analytically. The

critical points of the GB-G density function are the roots of the equation:

g′(x; ξ)g(x; ξ)

+g(x; ξ)

1−G(x; ξ)+

(c− 1) g(x; ξ){1−G(x; ξ)} [

log G(x; ξ)]−c (k+1)

g(x; ξ)[log G(x; ξ)

]c−1

G(x; ξ)[1 + {− log G(x; ξ)}] = 0.

(2.5)

The critical point of the hrf are obtained from the equation:

g′(x; ξ)g(x)

+g(x; ξ)

1−G(x; ξ)+

(c− 1) g(x; ξ){1−G(x; ξ)} [

log G(x; ξ)]−c

g(x; ξ)[log G(x; ξ)

]c−1

G(x; ξ)[1 + {− log G(x; ξ)}] = 0.

(2.6)

Note that there may be more than one root to Eqs. (2.5) and (2.6).

2.1.2 Infinite mixture representation

Here infinite mixture representation of the GBG density is presented.

Theorem 2.1.1. Let c > 0 and k > 0 are two real non-integer values. If X ∼ GBG(c, k, ξ),

then infinite mixture representations of the cdf and density are:

10


F (x) =∞∑

m=0

bm Hm(x), (2.7)

and

f(x) =∞∑

m=0

bm hm−1(x), (2.8)

where Hm(x) = Gm(x) and hm−1(x) represents the exp-G densities of the baseline distributions,

with m and m − 1 power parameters, respectively. The coefficients are given as b0 = 1 − a0 and

bm = −am where

am =∞∑

j=0

∞∑

m=0

m∑

i=0

(−1)cj+j+m+i

c j − icj

k + j − 1

j

m− c j

m

m

i

Pi,m.

(2.9)

Proof: If b > 0 is a real number, then the following series expansions

(1 + z)−b =∞∑

j=0

b + j − 1

j

(−1)j zj (2.10)

and

[log(1 + z)]a = a∞∑

k=0

k − a

k

k∑

i=0

(−1)k

a− i

k

i

Pi,k zk. (2.11)

where

Pj,k =1k

k∑

m=1

(jm− k + m)cmPj,k−m.

with pj,0 = 1 and ck = (−1)k

k+1

See (”http://functions.wolfram.com/ElementaryFunctions/Log/06/01/04/”)

Using (2.10), Eq. (2.2) becomes

F (x) = 1−∞∑

j=0

b + j − 1

j

(−1)j

{− log G(x)}c j (2.12)

11


Now, from (2.11), we obtain the Eq. (2.13) as

F (x) = 1−∞∑

j=0

∞∑

m=0

m∑

i=0

(−1)cj+j+i+m

c j − icj

b + j − 1

j

× m− c j

m

m

i

Pi,m Gm(x) (2.13)

The Eq. (2.13) can be expressed as

F (x) = 1−∞∑

m=0

am Hm(x),

The above equation can be written as

F (x) =∞∑

m=0

bm Hm(x).

where b0 = 1− a0 and bm = −am.

am are given in Eq. (2.9) and Hm(x) is the exp-G distribution of the baseline densities with

m as power parameter. We obtain Eq. (2.8) by simple derivation of Eq. (2.7).

2.1.3 Moments and moment generating function

The rth moments of the GBG family of distributions can be obtained as

E(Xr) =∞∑

m=0

bm

∞∫

0

xr hm−1(x) dx, (2.14)

where bm is defined in Eq.(2.9).

The sth incomplete moment of the GBG family of distributions can be obtained as

µs(x) =∞∑

m=0

bm T ′m(x), (2.15)

where T ′s(x) =x∫0

xs hm−1(x) dx.

The moment generating function of the GBG family of distributions can be defined by the

following expression as

MX(t) =∞∑

m=0

bm

∞∫

0

et x hm−1(x) dx. (2.16)

12


The mean deviations of the GBG family of distributions about the mean and median are,

respectively, defined as

Dµ = 2µF (µ)− 2µ1(µ), (2.17)

DM = µ− 2µ1(M), (2.18)

where µ = E(X) can be obtained from Eq.(2.14), M = Median(X) is the median can

be obtained from Eq. (2.4), F (µ) can be calculated easily from Eq. (2.2) and µ1(.) can be

obtained from Eq. (2.15). From the above equations, Bonferroni and Lorenz curves are

defined for a given probability π as

B(π) =µ1(q)π µ

L(π) =µ1(q)

µ, (2.19)

respectively. Here, q = F−1(π) is the GBG quantile function at π determined from Eq.(2.4).

2.1.4 Reliability parameter and Stochastic ordering

Reliability parameter

The expression for the reliability parameter is given by

R = P (X1 < X2) =

∞∫

0

f1(x, ξ1) F2(x, ξ2)dx

where X1 and X2 have independent GBG(c1, k1, ξ) and GBG(c2, k2, ξ) distributions with a

common parameter. Using the infinite mixture representations in Eqs. (2.7) and (2.8), we

obtain

R = P (X1 < X2) =∞∑

m=0

bm

∞∑

p=0

bp

∞∫

0

hm−1 Hp(x) dx,

where hm−1 and Hp(x) are the exp-G densities of the baseline distribution, with m− 1 and

p the power parameters.

Stochastic ordering

The concept of stochastic ordering are frequently used to show the ordering mechanism in

life time distributions. For more detail about stochastic ordering see (Shaked et al., 1994).

A random variable is said to be stochastically greater (X ≤st Y ) than Y if FX(x) ≤ FY (x)

for all x. In the similar way, X is said to be stochastically greater (X ≤st Y ) than Y in the

13


1. stochastic order (X ≤st Y ) if FX(x) ≥ FY (x) for all x,

2. hazard rate order (X ≤hr Y ) if hX(x) ≥ hY (x) for all x,

3. mean residual order (X ≤mrl Y ) if mX(x) ≥ mY (x) for all x,

4. likelihood ratio order (X ≤hr Y ) if fX(x) ≥ fY (x) for all x,

5. reversed hazard rate order (X ≤rhr Y ) if FX(x)FY (x) is decreasing for all x.

The stochastic orders defined above are related to each other, as the following implications.

X ≤rhr Y ⇐ X ≤lr Y ⇒ X ≤hr Y ⇒ X ≤st Y ⇒ X ≤mrl Y (2.20)

If X1 ∼ GBG(c, k1, ξ) and X2 ∼ GBG(c, k2, ξ) with c as the common parameter, then the

density functions of X1 and X2 are, respectively, given by

f(x) = c k1g(x)G(x)

{− log G(x)}c−1[1 + {− log G(x)}c

]k1−1,

g(x) = c k2g(x)G(x)

{− log G(x)}c−1[1 + {− log G(x)}c

]k2−1.

Then, their ratio will be

f(x)g(x)

=k1

k2

[1 + {− log G(x)}c

]k2−k1 .

Taking derivative with respect to x, we obtain

d

d x

f(x)g(x)

=k1

k2(k2 − k1)

c g(x) {− log G(x)}c−1

G(x) {log G(x)}[1 + {− log G(x)}c

]k2−k1−1.

From the above equation, we observe that if k1 < k2 then dd x

f(x)g(x) < 0, this implies that

likelihood ratio exists between X ≤lr Y .

2.2 Order Statistics

In this section, we give an explicit expression of the ith order statistics in terms of infinite

series of baseline densities.

14


Theorem 2.2.1. Let n be an integer value and X1, X2, ..., Xn, i = 1, 2, ..., n, be identically

independently distributed random variables. Then, the density function of ith order statistics is

given by

fi:n(x) =n−i∑

j=0

∞∑

m,r=0

mj(m, r) hm+r−1(x), (2.21)

where

mj(m, r) =n!m(−1)j bm ej+i−1:r

(i− 1)!j!(n− i− j)!(m + r)(2.22)

and hm+r−1(x) = (m + r) g(x) Gm+r−1(x) are the exp-G densities of the baseline distribution,

with m + r − 1 power parameter.

Proof:

If n ≥ 1 is an integer value then, we have following power series expansion (Gradshteyn

and Ryzhik, 2000)[ ∞∑

k=0

ak xk

]n

=∞∑

k=0

ak:n xk, (2.23)

where c0 = an0 and cm = 1

m a0

m∑k=1

(k n−m + k) ak cn:m−k.

The expression for ith order statistics is given by

fi:n(x) =n!

(i− 1)!(n− i)!g(x) Gi−1(x) [1−G(x)]n−i.

Using generalized binomial expansion, we obtain

fi:n(x) =n!

(i− 1)!(n− i)!

n−i∑

j=0

n− i

j

(−1)if(x)[F (x)]i+j−1.

Using the infinite mixture representations in Eqs.(2.7), (2.8) the Eq. (2.23) becomes

fi:n(x) =n−i∑

j=0

∞∑

r,m=0

mj(r,m)hr+m−1(x),

where the coefficients are given in Eq. (2.22).

15


2.3 Estimation of parameters

Here, the maximum likelihood estimates (MLEs) of the model parameters of the GBG fam-

ily complete and censored samples are studied. Let x1, x2, ..., xn be a random sample of

size n from the GBG family of distributions.

2.3.1 Estimation of parameters in case of complete samples

The log-likelihood function for complete samples for the vector of parameter Θ = (c, k, ξ)T

is given as

`(Θ) = n log(c k) +n∑

i=1

log g(xi; ξ)−n∑

i=1

log G(xi; ξ) + (c− 1)n∑

i=1

log{− log G(xi; ξ)}

− (k + 1)n∑

i=1

log[1 + {− log G(xi; ξ)}c

].

The components of the score vector U =(

∂l∂k , ∂l

∂c ,∂l∂ξ

)are given by

Uk =n

k−

n∑

i=1

log[1 + log G(xi; ξ)}c

],

Uc =n

c+

n∑

i=1

log{− log G(xi; ξ)} − (k + 1)n∑

i=1

[c {− log G(xi; ξ)}c log{− log G(xi; ξ)}

1 + log G(xi; ξ)}c

],

Uξ =n∑

i=1

[gξ(xi; ξ)g(xi; ξ)

]+

n∑

i=1

[Gξ(xi; ξ)G(xi; ξ)

]− (c− 1)

n∑

i=1

[Gξ(xi; ξ)[

1 + {− log G(xi; ξ)}c]

G(xi; ξ)

]

− c (k + 1)n∑

i=1

[c {− log G(xi; ξ)}c−1G(xi; ξ)

G(xi; ξ)[1 + {− log G(xi; ξ)}c

]]

.

Setting Uk, Uc and Uξ equal to zero and solving these equations simultaneously yields the

the maximum likelihood estimates.

2.3.2 Estimation of parameters in case of censored samples

Suppose that the lifetime of the first r failed items x1, x2, ..., xr have been observed. Then,

the likelihood function is given by

`(xi, Θ) =n!

(n− r)!

[r∏

i=1

f(xi; Θ)

]× (

F (x(0); Θ))n−r

, (2.24)

16


where f(.) and F (.) are the pdf and survival function corresponding to F (.), respectively.

Here, X = (x1, x2, ..., xr)T , Θ = (θ1, θ2, ..., θn)T and A is a constant. Inserting Eqs. (2.2)

and (2.3) into (2.24), we obtain

`(xi, Θ) = A

[r∏

i=1

c kg(xi; ξ)

1−G(xi; ξ){− log G(xi; ξ)

}c−1 (1 +{− log G(xi; ξ)

}c)−k−1

]

×[[

1 +{− log G(x(0); ξ)

}c]−k]n−r

. (2.25)

where A = n!(n−r)! .

Then, the log-likelihood function of parameters is given by

log `(xi, Θ) = log A + n log(c k) +r∑

i=1

log g(xi; ξ)−r∑

i=1

log G(xi; ξ)

+ (c− 1)r∑

i=1

log{− log G(xi; ξ)} − (k + 1)r∑

i=1


]

+ k (n− r) log[1 + {− log G(x(0); ξ)}c

].

The components of score vector U =(

∂l∂k , ∂l

∂c ,∂l∂ξ

)are given by

Uk =n

k−

r∑

i=1


]+ (n− r)

r∑

i=1

log[1 + {− log G(x(0); ξ)}c

],

Uc =n

c+

r∑

i=1

log{− log G(xi; ξ)} − (k + 1)r∑

i=1

[{− log G(xi; ξ)}c log{− log G(xi; ξ)}1 + {− log G(xi; ξ)}c

]

+ k (n− r)

[{− log G(x(0); ξ)}c log{− log G(x(0); ξ)}

1 + {− log G(x(0); ξ)}c

],

Uξ =r∑

i=1

[gξ(xi; ξ)g(xi; ξ)

]+

r∑

i=1

[Gξ(xi; ξ)

1−G(xi; ξ)

]+ (c− 1)

r∑

i=1

[Gξ(xi; ξ)

log G(xi; ξ) [1−G(xi; ξ)]

]

+ (k + 1)r∑

i=1

[c {− log G(xi; ξ)}c Gξ(xi; ξ)[

1 + {− log G(xi; ξ)}c]

[1−G(xi; ξ)]

]

− k(n− r)

[c {− log G(x(0); ξ)}c Gξ(x(0); ξ)[

1 + {− log G(x(0); ξ)}c] [

1−G(x(0); ξ)]]

.

Setting Uk, Uc and Uξ equal to zero and solving these equations simultaneously yields the


17


2.4 Special sub-models

In this section, we will give four special sub-models of the GBG family Viz, generalized

Burr Normal (GBN), generalized Burr Lomax (GBLx), generalized Burr exponentiated Ex-

ponential (GBEE) and generalized Burr exponentiated Uniform (GBU) distributions. For

illustration purpose, the GBU distribution is discussed in detail.

2.4.1 Generalized Burr Normal (GBN) distribution

Let the random variable X follows the normal distribution with the pdf g(x) = 1√2πσ

e−12(

x−µσ )2

and the cdf Φ(x) = 1√2πσ

∫ x−∞ e−

12(

x−µσ )2

dx, where µ > 0 and σ > 0 are scale and shape pa-

rameters, respectively and −∞ < x < ∞. Then, the cdf and pdf of the GB-N distribution

are given respectively,

F (x) = 1− [1 + {− log (1− Φ(x))}c]−k , (2.26)

and

f(x) =c k

σ2

µ− x

1− Φ(x){− log (1− Φ(x))}c−1 [1 + {− log (1− Φ(x))}c]−k−1 .

Here, X ∼ GBN(c, k, µ, σ) has a GB-N distribution. If c = 1, then GB-N distribution re-

duces to the generalized Lomax normal (GLxN) distribution. For k = 1, the GBN distribu-

tion reduces to the generalized log-logistic normal (GLLN) distribution. If c = k = 1, then

the GB-N distribution reduces to normal distribution. Figure 2.1(a) and Figure 2.1(b) gives

the plots for the density and hazard rate functions of the GB-N distribution, respectively.

For some values of the parameters. It is depicted by 2.1(b) that the failure rate function of

the GBN distribution can take decreasing and upside-down bathtub shapes for different

parametric combinations.

18


(a) (b)

−4 −2 0 2 4

0.0

0.1

0.2

0.3

0.4

0.5

µ = 0 σ = 1

x

pdf

c = 0.5 k = 4c = 2 k = 0.4c = 1 k = 1.5c = 1.8 k = 1

−4 −2 0 2 4

0.0

0.5

1.0

1.5

µ = 0 σ = 1

x

hrf

c = 0.5 k = 4c = 0.05 k = 15c = 0.2 k = 8c = 0.1 k = 5

Figure 2.1: Plots of (a) density and (b) hrf for the GBN distribution with different parameter

values.

2.4.2 Generalized Burr Lomax (GBLx) distribution

Let the random variable X follows the Lomax distribution having pdf g(x) = αβ (1 + αx)−β−1,

x > 0 and cdf G(x) = 1 − (1 + αx)−β , where α > 0 and β > 0 are scale and shape param-

eters, respectively, and 0 < x < ∞. The cdf and pdf of the GBLx distribution are given

respectively,

F (x) = 1− [1 + {β ln (1 + αx)}c]−k (2.27)

and

f(x) = c k α β (1 + αx)−1 {β ln (1 + αx)}c−1 [1 + {β ln (1 + αx)}c]−k−1 .

The random variable X ∼ GBLx(c, k, α, β) follows a GB-Lx distribution. If c = 1, then the

GBLx distribution reduces to generalized Lomax-Lomax (GLxLx) distribution. For k = 1,

then the GBLx distribution reduces to generalized log-logistic Lomax (GLLLx) distribu-

tion. If c = k = 1, then the GBLx distribution reduces to the Lomax distribution. For some

values of the parameters, the plots of the density and the failure rate function are shown

in Figure 2.2. It is depicted by 2.2(b) that the failure rate function of the GBLx distribution

can take increasing, decreasing and upside-down bathtub shapes for different parametric

combinations.

19


(a) (b)

0.0 0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

x

pdf

c = 0.3 k = 3 α = 0.8 β = 0.3c = 3 k = 0.8 α = 1 β = 2c = 5 k = 1.5 α = 2 β = 2c = 8 k = 2 α = 2 β = 1.5

0.0 0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

2.0

x

hrf

c = 0.3 k = 3 α = 0.8 β = 0.3c = 3 k = 0.8 α = 1 β = 2c = 5 k = 1 α = 1 β = 1c = 10 k = 2 α = 2 β = 1.3

Figure 2.2: Plots of the (a) density and (b) hrf for the GBLx distribution with different

parameter values.

2.4.3 Generalized Burr Exponentiated Exponential (GBEE) distribution

Let the random variable X follows the Exponentiated Exponential (EE) distribution having

pdf g(x) = α β e−α x (1− e−α x)β−1, x > 0 and cdf G(x) = (1− e−α x)β where α > 0

and β > 0 are shape parameters. Then cdf and pdf of the GB-EE distribution are given

respectively,

F (x) = 1−{

1 +[− log{1− (

1− e−α x)β}

]c}−k. (2.28)

and

f(x) =c k α β e−α x (1− e−α x)β−1

[− log{1− (1− e−α x)β}

]c−1

[1− (1− e−α x)β

] {1 +

[− log{1− (1− e−α x)β}

]c}k+1.

The random variable X ∼ GBEE(c, k, α, β) follows a GBEE distribution. If c = 1, then the

GBEE distribution reduces to the generalized Lomax-exponentiated exponential (GLxEE)

distribution. For k = 1, the GBEE distribution reduces to generalized exponentiated expo-

nential (GLLEE) distribution. If c = k = 1, then the GBEE distribution reduces to the EE

distribution. For some values of the parameters, the plots of the density and the failure rate

function are shown in Figure 2.3. It is depicted by 2.3(b) that the density function of the

GBEE distribution can take right-skewed, nearly symmetrical and reversed J shapes ,the

20


failure rate function of the GBEE distribution can take increasing, decreasing and upside-

down bathtub shapes for different parametric combinations.

(a) (b)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.0

0.5

1.0

1.5

x

pdf

c = 0.8 k = 3 α = 0.8 β = 0.3c = 2 k = 0.5 α = 2 β = 2c = 2 k = 5 α = 1 β = 5c = 5 k = 0.8 α = 0.9 β = 5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50.

00.

51.

01.

5

x

hrf

c = 0.9 k = 3 α = 0.8 β = 0.3c = 2 k = 0.5 α = 2 β = 2c = 1.5 k = 0.8 α = 2 β = 5c = 5 k = 0.9 α = 0.9 β = 8

Figure 2.3: Plots of the (a) density and (b) hrf for the GBEE distribution with different

parameter values.

2.4.4 Generalized Burr Uniform (GBU) distribution

Let the random variable X follows the Uniform distribution having the pdf g(x) = 1θ ,

x < θ and cdf G(x) = xθ , where θ > 0 is a scale parameter. Then, the cdf and pdf of the

GBU distribution are given respectively,

F (x) = 1−[1 +

{− ln

{1− x

θ

}}c]−k. (2.29)

and

f(x) =c k

θ − x

{− log

(1− x

θ

)}c−1 [1 +

{− log

(1− x

θ

)}c]−k−1.

The random variable X ∼ GBU(c, k, θ) follows a GBU distribution. If c = 1, then the GB-U

distribution reduces to generalized Lomax-uniform (GLxU) distribution while k = 1 the

GBU distribution reduces to generalized log-logistic uniform (GLLU) distribution. If c =

k = 1, then the GBU distribution reduces to Uniform distribution. For some values of the

parameters, the plots of the density and the failure rate function are shown in Figure 2.4.

21


It is depicted by 2.4(b) that the density function of the GBU distribution can take reversed

J shape, U-shape and J-shape, symmetrical, right-skewed shapes, the failure rate function

of the GBU distribution can take bathtub shapes for different parametric combinations.

(a) (b)

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

x

pdf

c = 2 k = 2 α = 5c = 3 k = 1 α = 5c = 2 k = 0.3 α = 5c = 0.5 k = 5 α = 5c = 1.5 k = 8 α = 5

0 1 2 3 4 5

0.0

0.5

1.0

1.5

2.0

x

hrf

c = 0.15 k = 15 α = 5c = 3 k = 0.8 α = 5c = 2 k = 1 α = 5c = 0.5 k = 5 α = 5c = 0.01 k = 25 α = 5

Figure 2.4: Plots of the (a) density and (b) hrf for the GBU distribution with different pa-

rameter values.

2.5 Mathematical properties of GBU distribution

The qf of the GBU distribution is given as

Qx(u) = θ

[1− e−{(1−u)−

1k−1} 1

c

].

The rth moment expression of the GBU distribution is given as

µ′r =∞∑

m=0

bm−1

(m

r + m

)θr. (2.30)

The sth incomplete moment of the GBU distribution is given as

T (m)s (z) =

∞∑

m=0

bm−1

( m

θm

) zm+s

m + s. (2.31)

The moment generating function of the GBU distribution as

MX(t) =∞∑

m=0

bm−1m(−1)m

θm

γ(m,−tx)tm

22


The first incomplete moment can be obtained by submitting s = 1 in Eq. (2.31) to get

T(m)1 (z) =

∞∑

m=0

bm−1

( m

θm

) zm+1

m + 1

The mean deviations about mean and median are, respectively, given by

D(µ) = 2µF (µ)− 2∞∑

m=0

bm−1

( m

θm

) µm+1

m + 1

D(M) = µ− 2∞∑

m=0

bm−1

( m

θm

) Mm+1

m + 1

The log-likelihood function of GBU is given by

`(Θ) = n log(

c k

θ

)−

n∑

i=1

log(1− xi

θ

)+ (c− 1)

n∑

i=1

log{− log

(1− xi

θ

)}

−(k + 1)n∑

i=1

log{

1 +{− log

(1− xi

θ

)}c}

where zi ={− log

(1− xi

θ

)}c.

The components of score vector are

Uk =n

k−

n∑

i=1

log (1 + zi),

Uc =n

c+

n∑

i=1

log{− log

(1− xi

θ

)}− (k + 1)

n∑

i=1

(zi:c

1 + zi

),

Uθ = n θ −n∑

i=1

[xi

θ(θ − xi)

]+ (c− 1)

n∑

i=1

[xi

θ(θ − xi) log(1− xi

θ

)]

+ (k + 1)n∑

i=1

(zi:θ

1 + zi

),

where zi:θ =−xi[− log (1−xi

θ )]c−1

θ(θ−xi)and zi:c =

[− log(1− xi

θ

)]c [log

{− log(1− xi

θ

)}].

These equations cannot be solved analytically and analytical softwares required to solve

them numerically.

2.5.1 Simulation and Application

Here, simulation and application on the GBU distribution is carried out.

23


2.5.2 Simulation

In this section, a simulation study is conducted to examine the performance of the MLEs of

the GBU parameters. We generate 1000 samples of size, n =20, 50, 100 and 500 of the GB-U

model. The evaluation of estimates was based on the mean of the MLEs of the model pa-

rameters, the mean squared error (MSE) of the MLEs. The empirical study was conducted

with software R and the results are given in Table 2.1. The values in Table 2.1 indicate that

the estimates are quite stable and, more importantly, are close to the true values for the

these sample sizes. It is observed from Table 2.1 that the standard deviation decreases as

n increases. The simulation study shows that the maximum likelihood method is appro-

priate for estimating the GB-U parameters. In fact, the MSEs of the estimated parameters

tend to be closer to the true parameter values when n increases. This fact supports that the

asymptotic normal distribution provides an adequate approximation to the finite sample

distribution of the MLEs. The normal approximation can be improved by using bias ad-

justments to these estimators. Approximations to the their biases in simple models may be

obtained analytically.

2.5.3 Applications

In this section, we use two real data sets to compare the fits of the GBG family with other

commonly used lifetime models. The parameters are estimated by the maximum likeli-

hood method using R language. First, the data sets are discribed and the MLEs and then

the corresponding standard errors (in parentheses) of the model parameters are given. The

values of the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC)

are also provided. Note that the lower the values of these criteria, the better the fit. For

both data sets, we use the sub-model GB-U to compare it with the Weibull-Uniform (WU),

Weibull-Burr XII (WBXII), beta-Burr XII (BBXII), and Kumaraswamy-Burr XII (KwBXII)

distributions.

24


Table 2.1: Mean and MSE for the of the MLEs of the parameters of the GB-U model.

c k θ n Mean MSE

c k θ c k θ

0.5 0.5 1 20 0.538 0.632 1.34 0.246 0.217 0.122

50 0.512 0.615 1.315 0.2407 0.155 0.103

100 0.5093 0.605 1.170 0.237 0.148 0.081

500 0.503 0.54 1.28 0.213 0.135 0.045

1 0.5 0.5 20 1.17 0.564 0.428 0.037 0.232 0.349

50 1.192 0.563 0.444 0.036 0.2205 0.3428

100 1.190 0.532 0.452 0.032 0.19 0.335

500 1.180 0.5192 0.478 0.0033 0.02 0.032

1 1 1 20 1.071 1.14 1.34 0.027 0.132 0.245

50 1.044 1.091 1.281 0.013 0.125 0.122

100 1.029 1.056 1.171 0.009 0.12 0.095

500 1.011 1.041 1.02 0.003 0.008 0.027

1 0.5 1 20 1.017 0.57 1.054 0.054 0.212 0.215

50 1.009 0.559 1.044 0.019 0.206 0.199

100 0.994 0.55 1.024 0.0095 0.194 0.035

500 0.998 0.538 1.024 0.001 0.184 0.0025

1 1 2 20 0.877 1.116 2.375 0.035 0.132 0.245

50 0.881 1.080 2.575 0.023 0.125 0.122

100 0.919 0.914 2.143 0.019 0.12 0.095

500 0.956 0.996 2.047 0.015 0.017 0.065

Data set 1: Birnbaum - Saunders data

The first data set was used by Birnbaum and Saunders (1969) and corresponds to the fa-

tigue time of 101 6061-T6 aluminum coupons cut parallel to the direction of rolling and

oscillated at 18 cycles per second (cps). Bourguignon et al. (2014) and Torabi and Montaz-

eri (2014) used Birnbaum and Saunders data.

It can be seen from Table 2.2 that AIC and BIC values of our model are the smallest among

all the other models. So, it is better than other models for this data set. Figure 2.5 shows

the estimated pdfs and cdfs of the fitted distributions, respectively. These graphs show a

25


good adjustment for the data of the estimated density, cumulative density functions of the

GB-U distribution.

Data set 2: Breast cancer data

The second real data set represents the survival times of 121 patients with breast cancer

obtained from a large hospital in a period from 1929 to 1938 (Lee, 1992). These data were

previously used by Ramos (2013), Tahir at el. (2015).

Table 2.3 indicate that the GBU model gives the best fit among all others competitive model

for breast cancer data. The estimated pdfs and cdfs are presented in Figure 2.6, respec-

tively. Figure 2.6 (a) also indicates that the GBU distribution provides a better fit to the

data than all other models.

Table 2.2: MLEs and their standard errors (in parentheses) for the data set 1.

Distribution MLE’s AIC BIC

GBU(c,k,θ) 5.848 1.075 215 - 910.211 915.421

(0.526) (0.116) - -

WU(a,b,θ) 1.184 2.782 300 - 945.090 950.301

(0.122) ( 0.181) - -

WBXII(c,k,α, β) 18.371 22.161 5.073 0.025 923.879 934.323

(116.891) (1.496) (1.048) (0.002)

BBXII(c,k,α, β) 66.295 51.750 0.815 31.009 916.923 927.343

(126.694) (38.003) (0.261) (44.806)

KwBXII(c,k,α, β) 793.469 588.060 20.290 0.048 915.017 925.438

( 219.976) (227.223) (15.357) (0.002)

26


(a) (b)

x

Den

sity

100 150 200

0.00

00.

005

0.01

00.

015

0.02

0

GBUWUWBXIIBBXIIKwXII

100 150 200

0.0

0.2

0.4

0.6

0.8

1.0

xcd

f


Figure 2.5: The estimated pdfs and cdfs of GBU and other competitive models for data set

1.

Table 2.3: MLEs and their standard errors (in parentheses) for data set 2.

Distribution MLE’s AIC BIC

GBU(c,k,θ) 1.183 3.505 160 - 1159.186 1164.777

(0.077) ( 0.340) - -

WU(a,b,θ) 1.405 0.691 160 - 1187.540 1193.132

(0.128) ( 0.042) - -

WBXII(c,k,α, β) 65.022 53.454 0.025 0.880 1166.053 1177.242

( 292.673) ( 9.906) (0. 011) ( 0.587)

BBXII(c,k,α, β) 0.418 159.033 0.366 28.783 1173.254 1184.437

(0.212) (104.782) (0.075) (10.892)

KwBXII(c,k,α, β) 32.582 341.059 0.169 1.687 1173.480 1184.663

(43.880) ( 293.180) ( 0.115) ( 1.602)

27


(a) (b)

x

Den

sity

0 50 100 150

0.00

00.

005

0.01

00.

015

0.02

0


0 50 100 150

0.0

0.2

0.4

0.6

0.8

1.0

x

cdf


Figure 2.6: The estimated pdfs and cdfs of GBU and other competitive models for data set

2.

2.6 Conclusion

We proposed a new family of distributions called Generalized Burr-G family of distribu-

tions. We studied most of its mathematical properties. Estimation of parameters are done

for both complete and censored samples, the mixture representation of ith order statistic

is given in terms of the baseline densities. Four special models are given and one of them

is discussed in detail. Simulation and application reveals that the proposed family gives

better results as compared to the competitive models.

28

Chapter 3

Marshall Olkin Burr G Family of

Distributions

3.1 Introduction

Marshall and Olkin proposed a flexible semi-parametric family of distribution and defined

new survival function (sf) as

F (x;α, ξ) =α G(x; ξ)

1− α G(x; ξ); α = 1− α. (3.1)

where α is an additional positive shape parameter. Clearly, α = 1 implies that F (x, α) =

G(x). The cdf, pdf and hrf corresponding to Eq. (3.1) are, respectively, given by

F (x;α, ξ) =G(x; ξ)

1− α G(x; ξ), (3.2)

f(x, α, ξ) =α g(x; ξ)(

1− α G(x; ξ))2 (3.3)

and

h(x, α, ξ) =α h(x; ξ)

1− α G(x; ξ). (3.4)

Recalling T-X family and considering the generator

W [G(x; ξ)] = − log[1−R(x; ξ)] = HR(x; ξ), (3.5)

29


where

∂

∂xW [G(x; ξ)] =

r(x; ξ)1−R(x; ξ)

= hR(x; ξ). (3.6)

Substituting Eqs. (3.5) and (3.6) BXII cdf defined in Eq. (1.1), we have

G(x; ξ) = 1− [1 + HR(x; ξ)c]−k . (3.7)

The pdf corresponding to (3.7), is given by

g(x; ξ) = c k hR(x; ξ) HR(x; ξ)c−1 [1 + HR(x; ξ)c]−k−1 . (3.8)

Substituting Eq. (3.7) in Eq. (3.1), the survival function of new family, that is Marshall

Olkin Burr-G for short(MOBG) family of distribution, is given by

F (x;α, c, k, ξ) =α

{1 +

[− log R(x; ξ)]c}−k

1− α{1 +


. (3.9)

The cdf and pdf corresponding to (3.9), are, respectively, given by

F (x;α, c, k, ξ) =1− {

1 +[− log R(x; ξ)

]c}−k

1− α{1 +


(3.10)

and

f(x; α, c, k, ξ) =c k r(x; ξ)

[− log R(x; ξ)]c−1 (

1 +[− log R(x; ξ)

]c)−k−1

[1−R(x; ξ)]{

1− α(1 +

[− log R(x; ξ)]c)−k

}2 . (3.11)

The qf can easily be obtained from Eq. (3.10) as

QX(u) = R−1

1− exp

−

{(u− 11− αu

)− 1k

− 1

} 1c

. (3.12)

Quantile function can be used to generate the data from the parent distribution, to obtain

median, skewness and kurkosis. Setting u = 0.5, median can be obtain

QX(0.5) = R−1

1− exp

−

{(0.5− 11− α0.5

)− 1k

− 1

} 1c

.

30


3.2 Infinite mixture representation

Here, infinite mixture representation of cdf and pdf of the MOBG family is obtained in

terms of the baseline cdf and pdf are given.

Theorem 3.2.1. If X ∼ MOBG(α, c, k), then we have following approximations. For α > 0

and c, k > 0 are the real non-integer values, then we have following mixture representation.

F (x) =∞∑

m=0

bm Hm(x), (3.13)

where Hm(x) = Gm(x) represents the exp-R distribution with power parameter m. The coefficients

are given as

am = α∞∑

j=0

αj∞∑

i=0

k(j + 1) + i− 1

i

(−1)i+c i c i

∞∑

m=0

m− c i

m

×m∑

l=0

(−1)l

c i− l

m

l

Pl,m (−1)m (3.14)

Eq. (3.13) shows that the density in (3.9) can be expressed as a infinite linear combination of the

baseline densities.

f(x) =∞∑

m=0

bm hm−1(x), (3.15)

where bm is defined in (3.14).

Proof:

If b > 0 is a real number, then we have following series expansions

(1− z)−b =∞∑

j=0

b + j − 1

j

zj (3.16)

(1− z)b =∞∑

j=0

b

j

(−1)j zj (3.17)

Let a > 1, then we have following log power series expansion

[log (1 + z)]a = a∞∑

k=0

k − a

k

k∑

j=0

(−1)j

a− j

k

j

Pj,kz

k, (3.18)

31


where

Pj,k =1k

k∑

m=1

(jm− k + m)cmPj,k−m.

with pj,0 = 1 and ck = (−1)k

k+1

(”http://functions.wolfram.com/ElementaryFunctions/Log/06/01/04/”)

From Eq. (3.17), we have Eq. (3.9)

F (x) = α∞∑

j=0

αj∞∑

i=0

k(j + 1) + i− 1

i

(−1)i+c i

[− log R(x)]c i (3.19)

Combining the results of Eq. (3.18) and Eq. (3.19), we get

F (x) = α∞∑

j=0

αj∞∑

i=0

k(j + 1) + i− 1

i

(−1)i+c i c i

∞∑

m=0

m− c i

m

×m∑

l=0

(−1)l

c i− l

m

l

Pl,m (−1)mRm(x)

The cdf can be obtained from the above equation. In simplified form, we have

F (x) = 1−∞∑

m=0

am Hm(x)

It can be written as

F (x) =∞∑

m=0

bm Hm(x),

where b0 = 1 − a0, bm = −am and coefficients am are given in Eq. (3.14) and Hq(x) is the

exp-G distribution of the base line densities with q as power parameter. Eq. (3.15) can be

obtained by simple derivation of Eq. (3.13).

32


3.3 Asymptotics and Shapes

If x → 0, then the asymptotic of pdf, cdf and hrf are given by:

f(x) ∼ c k r(x)

{− log R(x)}c−1

α2,

F (x) ∼1− [

1 +{− log R(x)

}c]

α,

h(x) ∼ c k r(x)

{− log R(x)}c−1

α.

If x →∞, then the asymptotic of pdf, cdf and hrf are given by:

f(x) ∼ c k r(x)

{− log R(x)}c−1

1−R(x),

F (x) ∼ 1,

h(x) ∼c k r(x)

{− log R(x)}c−1

{1−R(x)} [1 +

{− log R(x)}c] ,

The shapes of the density and hazard rate functions of MOBG can be defined analyti-

cally. The critical points of the MOBG density function are the roots of the equation:

r′(x; ξ)r(x; ξ)

+r(x; ξ)

1−R(x; ξ)+

(c− 1)r(x; ξ) {1−R(x; ξ)}−1

{− log R(x; ξ)} +

(k + 1)c r(x; ξ){− log R(x; ξ)

}c−1

{1−R(x; ξ)} [1 +

{− log R(x; ξ)}c]

−2c k α r(x; ξ)

{− log R(x; ξ)}c−1 [

1 +{− log R(x; ξ)

}c]−k−1

{1−R(x; ξ)}[1− α (1 + (HR(x; ξ))c)−k

] = 0.

This equation may have more than one root.

The critical point of hazard rate function of MOBG family are the roots of the equation:

r′(x; ξ)r(x; ξ)

+r(x; ξ)

1−R(x; ξ)+

(c− 1) r(x; ξ) {1−R(x; ξ)}−1

{− log R(x; ξ)} +

c r(x; ξ){− log R(x; ξ)

}c−1

{1−R(x; ξ)} [1 +

{− log R(x; ξ)}c]

c k α r(x; ξ){− log R(x; ξ)

}c−1 [1 +

{− log R(x; ξ)}c]−k−1

{1−R(x; ξ)}[1− α (1 + (HR(x; ξ))c)−k

] = 0.

3.4 General properties

Here, the general properties of MOBG are obtained mathematically Viz. rth moment, sth

incomplete moment, mgf and mean deviations.

33


3.4.1 Moments

The rth moment of the MOBG family of distributions can be obtained by using the follow-

ing expression

E(Xr) =∞∑

m=0

bm

∞∫

0

xr hm−1(x) dx, (3.20)

where bm is defined in Eq. (3.14), hm−1(x) = mr(x) Rm−1(x) and m − 1 is the power

parameter.

Similarly, the sth incomplete moment of the MOBG family of distributions can be obtained

as

µs(x) =∞∑

m=0

bm T ′s(x), (3.21)

where T ′s(x) =x∫0

xs hm−1(x)dx.

The moment generating function of the MOBG family of distributions is obtained as

MX(t) =∞∑

m=0

bm Mm−1(t), (3.22)

where Mm−1(t) =∞∫0

et x hm−1(x)dx.

The mean deviations of the MOBG family of distributions about the mean and median,

can be obtained from the relations

Dµ = 2µF (µ)− 2µ1(µ) (3.23)

and

DM = µ− 2µ1(M) (3.24)

where µ = E(X), can be obtained from Eq. (3.20), M = Median(X), is the median given

in Eq. (3.12), F (µ) can be calculated from Eq. (3.10) and µ1(.) can be obtained from Eq.

(3.21). Other applications of the equations above are obtaining the Bonferroni and Lorenz

curves defined for a given probability π as

B(π) =µ1(q)π µ

and L(π) =µ1(q)

µ(3.25)

where q = F−1(π), is the MOBG quantile function at π.

34


3.4.2 The Stress-Strength reliability parameters

The reliability parameter R, when X1 and X2 have independent MOBG(c1, k1, α1) and

MOBG(c2, k2, α2) distributions with the common shape parameter and scale parameter

can be obtained from Eqs. (3.10) and (3.11)

R = P (X1 < X2) =

∞∫

0

f1(x) F2(x)dx. (3.26)

Using the infinite mixture representation given in Eqs. (3.13) and (3.15), we have

R = P (X1 < X2) =∞∑

m=0

∞∑

p=0

ap bm

∞∫

0

hp−1(x) Hm(x)dx, (3.27)

where Hm(x) = Rm(x) and hm−1(x) = mr(x) Rm−1(x) are the exp-R densities of the

baseline distribution.

3.4.3 Stochastic ordering

If X1 ∼ MOBG(c, k, α1) and X2 ∼ MOBG(c, k, α2) with c and k as the common parame-

ter, then the density functions of X1 and X2 are, respectively, given by

f(x) =α1 bc,k(x)

{1− α1 [1−Bc,k(x)]}2

and

g(x) =α2 bc,k(x)

{1− α2 [1−Bc,k(x)]}2

Then their ratio will be

f(x)g(x)

=α1

α2

[1− α2 [1−Bc,k(x)]1− α1 [1−Bc,k(x)]

]2

Taking derivative of the above ratio with respect to x, we get

d

d x

f(x)g(x)

= 2α1

α2

[(α2 − α1) bc,k(x) {1− α2 [1−Bc,k(x)]}

{1− α1 [1−Bc,k(x)]}2

]

From the above equation, we observe that, if α1 < α2 ⇒ ddx

f(x)g(x) < 0, then this implies that


35



Here, the expression of the ith order statistics is defined as a infinite series of baseline

densities.

Theorem 3.5.1. If n is an integer value and for i = 1, 2, ..., n and X1, X2, ..., Xn be identically

independently distributed random variables. Then the density of ith order statistics is

fi:n(x) =n−i∑

j=0

∞∑

r,m=0

Vj(r,m) hr+m−1(x), (3.28)

where hr+m−1(x) = (r + m) r(x) Rr+m−1(x) are the exp-G densities of the baseline distribution,

with power parameter r + m− 1 and the coefficients are given by

Vj(r,m) =n! (−1)j br em:j+i−1 r

(i− 1)!j!(n− i− j)!(r + m)(3.29)

Proof:

If n ≥ 1 is an integer value then, we have following power series expansion (Gradshtegn

and Ryzhik, 2000).( ∞∑

k=0

ak xk

)n

=∞∑

k=0

ak:n xk, (3.30)


m a0

m∑k=1


The expression for ith order statistics is defined as

fi:n(x) =n!

(i− 1)!(n− i)!g(x) Gi−1(x) [1−G(x)]n−i

Using the series expansion in Eq. (3.17), we get

fi:n(x) =n!

(i− 1)!(n− i)!

n−i∑

j=0

n− i

j

(−1)if(x)[F (x)]i+j−1.

Using the infinite mixture representation of MOBG densities in Eqs. (3.13), (3.15) and

(3.30), we get

fi:n(x) =n−i∑

j=0

∞∑

r,m=0

Vj(r,m) hr+m−1(x),

where Vj(r,m) are defined in Eq. (3.29).

36


3.6 Estimation of parameters

Here, the maximum likelihood estimates (MLEs) of the model parameters of the MOBG

family complete and censored samples are given. Let x1, x2, ..., xn be a random sample of

size n from the MOBG family of distributions.


The log-likelihood function for the vector of parameter Θ = (α, c, k, ξ)T is

l(Θ) = n log(α c k) +n∑

i=1

log r(xi)−n∑

i=1

log R(xi) + (c− 1)n∑

i=1

log{− log R(xi)}

−(k + 1)n∑

i=1

log[1 + {− log R(xi)}

]− 2n∑

i=1

log{

1− α[1 + {− log R(xi)}c]−k}


Uα =n

α+ 2

n∑

i=1

[[1 + {− log R(xi)}c]−k

{1− α [1 + {− log R(xi)}c]−k

}]

,

Uk =n

k− 2

n∑

i=1

[α [1 + {− log R(xi)}c]−k log

{1 + {− log R(xi)}c

}{1− α [1 + {− log R(xi)}c]−k

}]

,

−n∑

i=1

log[1 + {− log R(xi)}c

]

Uc =n

c+

n∑

i=1

log{− log R(xi)} − (k + 1)n∑

i=1

[{− log R(xi)}c log

{− log R(xi)}

1 + {− log R(xi)}c

],

−2n∑

i=1

[α k [1 + {− log R(xi)}c]−k−1 {− log R(xi)}c log

{− log R(xi)}

1− α [1 + {− log R(xi)}c]−k

].

Setting Uk, Uc and Uα equal to zero and solving these equations simultaneously yields the


3.6.2 Estimation of parameters in case of censored complete samples


the likelihood function is given by

L =n!

(n− r)!

[r∏

i=1

f(xi; Θ)

]× (

F (x(0); Θ))n−r

, (3.31)

37


where f(.) and F (.) are the pdf and survival function corresponding to F (.), respectively.

Here, X = (x1, x2, ..., xr)T , Θ = (θ1, θ2, ..., θn)T . If r = n equation (3.31) turns out to be

likelihood function for complete samples. Submitting the equations (3.11) and (3.10) in

equation (3.31) the Log likelihood function is

log L = logn!

(n− r)!+ r log c + r log k +

r∑

i=1

log r(x(i)) + (c− 1)r∑

i=1

log{− log R(i)

}

− (k + 1)r∑

i=1

log[1 +

{− log R(i)

}c]− 2r∑

i=1

log[1− α

[1 +

{− log R(i)

}c]−k]

+ log α− k log[1 +

{− log R(0)

}c]− log[1− α

[1 +

{− log R(0)

}c]−k]−

r∑

i=1

log R(i)


Uk =r

k−

r∑

i=1

log[1 +

{− log R(i)

}c]− 2 αr∑

i=1

[1 +

{− log R(i)

}c]−k log[1 +

{− log R(i)

}c][1− α

[1 +

{− log R(i)

}c]−k]

− (n− r) log[1 +

{− log R(0)

}c]− αr∑

i=1

[1 +

{− log R(0)

}c]−k log[1 +

{− log R(0)

}c][1− α

[1 +

{− log R(0)

}c]−k] ,

Uc =r

c+

r∑

i=1

log{− log R(i)

}− (k + 1)r∑

i=1

{− log R(i)

}c log[{− log R(i)

}]

1 +{− log R(i)

}c

− 2 α kr∑

i=1

{− log R(i)

}c log{− log R(i)

} [1 +

{− log R(i)

}c]−k−1

[1− α

[1 +

{− log R(i)

}c]−k]

− (n− r) α k{− log R(0)

}c log{− log R(0)

} [1 +

{− log R(0)

}c]−k−1

[1− α

[1 +

{− log R(0)

}c]−k]

− k{− log R(0)

} log{− log R(0)

}[1 +

{− log R(0)

}c] ,

Uα = −2

[1 +

{− log R(i)

}c]−k

[1− α

[1 +

{− log R(i)

}c]−k] + (n− r)

1

α−

[1 +

{− log R(0)

}c]−k

[1− α

[1 +

{− log R(0)

}c]−k] .

Setting Uk, Uc and Uα equal to zero and solving these equations simultaneously yields

the the maximum likelihood estimates.

38


3.7 Special models of MOBG family

In this section, four special models of MOBG family are discussed Viz. MOB Frechet

(MOBFr), MOB log logistic (MOBLL), MOB Weibull (MOBW) and MOB Lomax (MOBLx)

distributions. The density and hazard rate Plots for some parameters are displayed to

illustrate the flexibility of these distributions.

3.7.1 Marshall-Olkin Burr XII Frechet (MOBFr) distribution

Let a random variable X follows the Frechet distribution as baseline distribution with

pdf and cdf r(x) = a bx2

(ax

)b−1e−( a

x)b

, x ≥ 0 and R(x) = e−( ax)b

. where a > 0 and

b > 0 respectively, are the scale and shape parameters. Then the cdf and pdf of MOBFr-

distribution are, respectively, given by

F (x) =α

{1 +

[− log(1− e−( a

x)b

)]c}−k

1− α

{1 +

[− log(1− e−( a

x)b

)]c}−k

(3.32)

and

f(x) =c k α

(ax

)b−1e−( a

x)b[− log(1− e−( a

x)b

)]c−1 {

1 +[− log(1− e−( a

x)b

)]c}−k−1

x2

(1− e−( a

x)b) [

1− α

{1 +

[− log(1− e−( a

x)b

)]c}−k

]2 .

(i) If α = 1, then MOBFr distribution reduces to OBFr distribution, (ii) if α = c = 1,

then MOBFr distribution reduces to OLxFr distribution, (iii) if α = k = 1, then MOBFr

distribution reduces to OLLFr distribution, (iv) if α = c = k = 1, then MOBFr distribution

reduces to Frechet distribution. Figure 3.1, shows the plots of density and hazard rate

functions of MOBFr distribution. The density of MOBFr gives symmetrical, right-skewed

and reversed-J shapes. While hrf gives decreasing and upside-down bathtub shapes.

3.7.2 Marshall-Olkin Burr XII log-logistic (MOBLL) distribution

Let a random variable X follows the log-logistic distribution as baseline distribution with

pdf and cdf r(x) = θλ

(xλ

)θ−1(1 +

(xλ

)θ)−2

, x ≥ 0 and R(x) = 1 −(1 +

(xλ

)θ)−1

, where

39


(a) (b)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.5

1.0

1.5

2.0

x

pdf

c = 0.5 k = 1.5 λ = 0.5 α = 1.2c = 1.5 k = 2 λ = 1.5 α = 1.3c = 2.5 k = 1.5 λ = 2 α = 1.3c = 4 k = 2 λ = 2 α = 3

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

x

hrf

c = 0.5 k = 1.5 λ = 0.5 α = 1.2c = 1.5 k = 1.5 λ = 1.5 α = 1.3c = 2.5 k = 0.5 λ = 2 α = 1.3c = 0.3 k = 0.8 λ = 1.8 α = 3

Figure 3.1: Plots of (a) density and (b) hrf for MOBFr distribution for different parametric

values.

a > 0 and b > 0 respectively, are the scale and shape parameters. Then the cdf and pdf of

MOBLL-distribution are, respectively, given by

F (x) =1−

{1 +

[log

(1 +

(xλ

)θ)]c}−k

1− α{

1 +[log

(1 +

(xλ

)θ)]c}−k

(3.33)

and

f(x) =c k α θ

(xλ

)θ−1[log

(1 +

(xλ

)θ)]c−1 {

1 +[log

(1 +

(xλ

)θ)]c}−k−1

λ(1 +

(xλ

)θ) [

1− α{

1 +[log

(1 +

(xλ

)θ)]c}−k

]2

(i) If α = 1, then MOBLL distribution reduces to OBLL distribution, (ii)if α = c = 1,

then MOBLL distribution reduces to OLxLL distribution, (iii)if α = k = 1, then MOBLL

distribution reduces to OLLLL distribution, (iv) if α = c = k = 1, then MOBLL distribution

reduces to Log logistic distribution. Figure 3.2 gives the plots of density and hazard rate

functions of MOBLL distribution. The density shapes of MOBLL are symmetrical, right-

skewed and reversed-J shapes. The hrf of MOBLL are increasing, decreasing and upside-

down bathtub shapes.

40


(a) (b)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.5

1.0

1.5

x

pdf

c = 0.5 k = 0.5 λ = 1.5 θ = 1 α = 0.5c = 1.5 k = 2 λ = 1.5 θ = 1 α = 1.3c = 14 k = 2 λ = 1.5 θ = 1 α = 2c = 2.5 k = 1.5 λ = 1 θ = 1 α = 2

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

x

hrf

c = 0.5 k = 0.5 λ = 1.5 θ = 1 α = 0.5c = 1.5 k = 2 λ = 1.5 θ = 1 α = 1.3c = 2.5 k = 1.5 λ = 1 θ = 1 α = 1.3c = 4 k = 2 λ = 2 θ = 2 α = 3

Figure 3.2: Plots of (a) density and (b) hrf for MOBLL distribution for different parametric

values.

3.7.3 Marshall-Olkin Burr XII-Weibull (MOBW) distribution

Let a random variable X follows the Weibull distribution as baseline distribution with pdf

and cdf r(x) = a b xb−1 e−a xb, x ≥ 0 and R(x) = 1− e−a xb

, where a > 0 and b > 0 respec-

tively, are the scale and shape parameters. Then the cdf and pdf of MOBW-distribution

are, respectively, given by

F (x) =1− [

1 +(axb

)c]−k

1− α[1 + (axb)c]−k

(3.34)

and

f(x) =c k a b α xb−1

(axb

)c−1 [1 +

(axb

)c]−k−1

[1− α

[1 + (axb)c]−k

]2

(i) If α = 1, then MOBW distribution reduces to OBW distribution, (ii) if α = c = 1,

then MOBW distribution reduces to OLxW distribution, (iii) if α = k = 1, then MOBW

distribution reduces to OLLW distribution, (iv) if α = c = k = 1, then MOBW distribution

reduces to Log logistic distribution. Figure 3.3 shows the plots of density and hazard rate

functions of MOBW distribution. The pdf gives symmetrical, right-skewed and reversed-J

shapes. The hrf of MOBW are increasing, decreasing and upside-down bathtub shapes.

41


(a) (b)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.5

1.0

1.5

x

pdf

c = 1.5 k = 1 λ = 3 θ = 1 α = 5c = 0.8 k = 0.8 λ = 3 θ = 1 α = 0.5c = 0.5 k = 2 λ = 2 θ = 1 α = 0.5c = 3.5 k = 1.5 λ = 2 θ = 0.5 α = 2

0.0 0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

2.0

x

hrf

c = 0.1 k = 1 λ = 3 θ = 1 α = 0.5c = 0.5 k = 0.8 λ = 3 θ = 1 α = 0.5c = 0.5 k = 1.5 λ = 2 θ = 1 α = 0.5c = 3 k = 1 λ = 2.2 θ = 0.5 α = 4c = 1 k = 1 λ = 2 θ = 1.5 α = 4

Figure 3.3: Plots of (a) density and (b) hrf for MOBW distribution for different parametric

values.

3.7.4 Marshall-Olkin Burr XII Lomax (MOBLx) distribution

Let a random variable X follows the Lomax distribution as base line distribution with pdf

and cdf r(x) = ba

(1 + x

a

)−b−1, x ≥ 0 and R(x) = 1 − (

1 + xa

)−b, where a > 0 and b > 0

respectively, are the shape and scale parameters. Then cdf and pdf of MOBLx-distribution


F (x) =1− [

1 +(b log

(1 + x

a

))c]−k

1− α[1 +

(b log

(1 + x

a

))c]−k(3.35)

and

f(x) =b c k α

(b log

(1 + x

a

))c−1 [1 +

(b log

(1 + x

a

))c]−k−1

a(1 + x

a

) [1− α

[1 +

(b log

(1 + x

a

))c]−k]2 .

(i) If α = 1, then MOBLx distribution reduces to OBLx distribution, (ii)if α = c = 1,

then MOBLx distribution reduces to OLxLx distribution, (iii)if α = k = 1, then MOBLx

distribution reduces to OLLLx distribution, (iv) if α = c = k = 1, then MOBLx distribution

reduces to log logistic distribution. Figure 3.4 gives the plots of density and hazard rate

functions of MOBLx distribution. The shapes of density of MOBLx are right-skewed and

reversed-J shapes. The hrf of MOBLx are increasing, decreasing and upside-down bathtub

shapes.

42


(a) (b)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

x

pdf

c = 0.5 k = 2 λ = 1.5 θ = 1 α = 0.5c = 1.8 k = 2 λ = 1 θ = 1 α = 1.5c = 1.5 k = 2 λ = 1.5 θ = 1 α = 2.5c = 1.5 k = 2 λ = 1.5 θ = 1 α = 0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

x

hrf

c = 0.5 k = 2 λ = 1.5 θ = 1 α = 0.5c = 0.8 k = 1.5 λ = 1.5 θ = 1 α = 0.5c = 1.5 k = 2 λ = 1.5 θ = 1 α = 0.3c = 1.5 k = 2 λ = 1.5 θ = 1 α = 1.5c = 5 k = 0.5 λ = 0.3 θ = 0.5 α = 2.5

Figure 3.4: Plots of (a) density and (b) hrf for MOBLx distribution for different parametric

values.

3.8 Mathematical properties of MOBLx distribution

In this section, the properties of MOBLx distribution are briefly discribed.

From Eqs.(3.13) and (3.15), the cdf and pdf of the density of MOBLx distribution can be

expressed in terms of infinite mixture form as

F (x) =∞∑

m=0

bm

{1−

(1 +

x

a

)−b}m

.

and

f(x) =∞∑

m=0

bm(m)b

a

(1 +

x

a

)−b−1{

1−(1 +

x

a

)−b}m−1

.

The qf of MOBLx distribution can be obtained by inverting from Eq. (3.12) as.

Qx(u) = a[(1−A)−

1b − 1

], (3.36)

where A = 1− exp

[−

{(u−11−α

)− 1k − 1

} 1c

].

From Eq. (3.20) the rth moment expression of MOBLx distribution will be

µ′r =∞∑

m=0

bm marb∞∑

j=0

m− 1

j

(−1)jB(r + 1, β(j + 1)− r).

43


From Eq. (3.21) the sth incomplete moment of MOBLx distribution will be

ms =∞∑

j=0

vj,masbBxa(s + 1, β(j + 1)− s),

where vj,m =∞∑

m=0bm m

m− 1

j

(−1)j .

The first incomplete moment of MOBLx distribution can be obtain by setting s = 1 in the

above expression

m1 =∞∑

j=0

vj,ma b Bxa(2, β(j + 1)− 1). (3.37)

From Eq. (3.22) the moment generating function of MOBLx distribution will be

MX(t) =∞∑

i=0

vi,j,m Γ(i + 1)(−1

t

)i+1

,

where vi,j,m =∞∑

m,j=0bm m

m− 1

j

β(j + 1) + i

i

(−1)i+j .

The mean deviations about mean and median of MOBLx distribution can easily be ob-

tained from Eqs. (3.23) and (3.24), δ1 = 2µ′1F (µ′1)− 2mµ′1 and δ2 = µ′1 − 2mM .

where F (µ′) can be obtained form Eq. (3.10), median for MOBLx distribution can be ob-

tained from Eq. (3.36) by setting u = 0.5 and the first incomplete moment is given in Eq.

(3.37).

Let a is the common parameter between two MOBLx distributions such as

MOBLx(α1, c1, k1, a, b1) and MOBLx(α2, c2, k2, a, b2). Then from Eq. (3.27), the reliability

parameter for MOBLx distribution is

R =∞∑

m=0

∞∑

p=0

bp bm

p− 1

i

m

j

(−1)i+j p b2

{b2 (i + 1) + b1 j} .

Let x1, ..., xn be a sample of size n from the MOBLx distribution, then the log-likelihood

function for the vector of parameters can be expressed as

l(Θ) = log{

c k b α

a

}−

n∑

i=1

log(1 +

x

a

)+ (c− 1)

n∑

i=1

log{

b log(1 +

x

a

)}

− (k + 1)n∑

i=1

log(1 + B)− 2n∑

i=1

log[1− α(1 + B)−k

],

44


where B ={b log

(1 + x

a

)}c

The components of score vector are.

Uα =n

α−

n∑

i=1

(1 + B)−k

1− α(1 + B)−k.

Uk =n

k−

n∑

i=1

log(1 + B)− 2n∑

i=1

α (1 + B)−k log(1 + B)1− α(1 + B)−k

.

Uc =n

c+

n∑

i=1

log{

b log(1 +

x

a

)}− (k + 1)

n∑

i=1

Bi:c

1 + B− 2

n∑

i=1

α k (1 + B)−k−1 B′i:c

1− α(1 + B)−k.

Ub =n

b+ n

(c− 1)b

− (k + 1)n∑

i=1

B′i:b

1 + B− 2

n∑

i=1

α k (1 + B)−k−1 B′i:b

1− α(1 + B)−k.

Ua = n a +n∑

i=1

x

a2(1 + x

a

) − (c− 1)n∑

i=1

x

a2(1 + x

a

) {log

(1 + x

a

)} − (k + 1)n∑

i=1

B′i:a

1 + B

− 2n∑

i=1

α k (1 + B)−k−1 B′i:a

1− α(1 + B)−k.

From Eq. (3.28) the density of the ith order statistic of MOBLx distribution can be written

as

fi:n(x) =∞∑

j=0

∞∑

r,m=0

Vj(r,m)(1 +

x

a

)−b−1{

1−(1 +

x

a

)−b}m+r−1

.

where the coefficients are defined in Eq. (3.29).

3.8.1 Simulation study of MOBLx distribution

In this section, simulation is carried out to access the performance of ML estimates of the

MOBLx distribution of different sizes (n=50, 150, 300). 500 samples are simulated for the

true parameters values I: a= 3 b= 4.5 c= 0.5 k= 4 α= 2 and II : a= 0.2 b= 0.8 c= 1.5 k= 8

α= 7 in order to obtain average estimates (AEs), biases and mean square errors (MSEs)

of the parameters. which are listed in Table 3.1. The small values of the biases and MSEs

indicate that the maximum likelihood method performs quite well in estimating the model

parameters of the MOBLx distribution.

45


Table 3.1: Estimated AEs, biases and MSEs of the MLEs of parameters of MOBLx distribu-

tion based on 500 simulations of with n=50, 100 and 300.

I II

n parameters A.E Bias MSE A.E Bias MSE

50 a 2.251 0.749 1.763 0.224 0.024 0.010

b 5.010 0.510 0.351 0.885 0.085 0.076

c 0.735 0.235 0.072 2.222 0.722 0.541

k 4.847 0.847 1.934 8.016 0.016 0.006

α 2.593 2.407 1.240 6.989 0.011 1.038

150 a 2.549 0.451 0.651 0.157 0.020 0.004

b 4.953 0.453 0.244 0.708 0.072 0.063

c 0.776 0.206 0.062 2.218 0.718 0.531

k 4.631 0.631 1.113 7.980 0.012 0.002

α 1.918 1.082 0.851 6.985 0.009 0.961

300 a 2.368 0.332 0.539 0.184 0.016 0.003

b 4.996 0.396 0.210 0.767 0.033 0.007

c 0.754 0.154 0.052 2.177 0.677 0.050

k 4.689 0.589 1.106 7.984 0.010 0.001

α 2.062 0.938 0.818 7.003 0.003 0.910

3.8.2 Application

In this section, the performance of the MOBG family is assessed by considering a spe-

cial model MOBLx model through two real life data sets. The MOBLx model is com-

pared with existing models: generalized exponentiated exponential Weibull (GEEW), Ku-

maraswamy Lomax (KLx), Beta Lomax (BLx), Lomax (Lx), generalized exponentiated ex-

ponential (GEE) and exponentiated-Weibull (EW) distributions. The maximum likelihood

method is used to estimate the model parameters and their standard errors. The model

adequacy measures such as, Anderson Darling (A*), Cramer von Mises goodness (W*) are

used to compare these models.

3.8.3 Data set 3: Carbon Fibres data

The data set discribeed the breaking stress of carbon Fibres (in Gba) used earlier by Cordeiro

et al.(2013).

46


3.8.4 Data set 4: Remission Times data

The data set represents the remission times (in months) of a random sample of 128 bladder

cancer patients was reported by Lee et al. (2003).

Table 3.2: The parameter estimates and A* and W* values for data set 3

Distribution c k α a b A* W*

MOBLx 1.92 33.3 20.99 18.83 2.15 0.2636 0.04242

(1.25) (109.8) (47.87) (75.26) (9.46)

GEEW 0.15704 0.03692 3.22861 1.77021 - 0.37840 0.05954

(0.37787) (0.03898) (0.63676) (1.38506)

KLx 103.18 8.72 - 3.90 345.35 0.5807 0.1059

(31.22) (26.57) - (0.603) (72.11)

BLx 181.89 7.02 - 7.57 68.44 1.339 0.2474

(38.46) (40.64) - (1.30) (38.33)

Lx 109.20 39.67 - - - 1.364 0.2516

(19.55) (12.807)

GEE 0.26555 10.0365 7.23658 - - 1.43415 0.26682

(0.21621) (2.59504) (7.05288)

EW 3.73666 0.01709 0.01402 - - 0.40365 0.06479

(0.44575) (0.02134) (0.00845)

3.9 Concluding remarks

In this chapter, a family of distributions called ”Marshall Olkin Burr-G family of distri-

butions” is proposed. Most of the mathematical properties of this family are studied in-

cluding qf, infinite mixture representation MOBG densities, rth moment, sth incomplete

moment, moment generating function, mean deviations, reliability parameter are stud-

ied. Expression for ith order statistics is given and estimation of parameters are done by

Maximum likelihood method for complete and censored samples. A special sub-model

is discussed in detail for illustration propose. Finally application is carried out on two

real data set to check the performance of the proposed family which provides consistently

better fit than other competitive models.

47


(a) (b)

x

Den

sity

0 1 2 3 4 5

0.0

0.1

0.2

0.3

0.4

0.5

MOB−LLoB−LK−L

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

x

cdf

MOB−LLoB−LK−L

(c) (d)

x

Den

sity

0 20 40 60 80

0.00

0.02

0.04

0.06

0.08

0.10 MOB−L

LoB−LK−L

0 20 40 60 80

0.0

0.2

0.4

0.6

0.8

1.0

x

cdf

MOB−LLoB−LK−L

Figure 3.5: Plots of estimated pdf (a) and (c), cdf (b) and (d) for data set 3 and data set 4.

48


Table 3.3: The parameter estimates and A* and W* values for data set 4

Distribution c k α a b A* W*

MOBLx 1.64953 0.08757 1.15492 32.19600 21.31120 0.09018 0.01391

(6.0144) (0.1735) (0.8478) (58.6221) (61.8282)

GEEW 1× 10−10 1.30988 0.52009 3.74791 - 0.29907 0.04526

(0.098282) (1.91117) (0.3223) (3.39406)

KLx 13.19 0.539 - 1.518 8.289 0.1724 0.0258

(17.68) (2.712) - (0.2667) (47.47)

BLx 20.63 0.0867 - 1.585 54.60 0.1923 0.0286

(14.18) (0.3135) - (0.2836) (19.93)

Lx 121.041 13.94 - - - 0.4873 0.0806

(42.76) (15.39)

GEE 0.12117 1.21795 1.00156 - - 0.71819 0.12840

(0.1068) (0.1877) (0.8659)

EW 1.04783 1.005× 10−7 0.09389 - - 0.96345 0.15430

(0.31424) (0.3013) (0.1179)

49

Chapter 4

Odd Burr-G Poisson Family of

distributions

4.1 Introduction

In this chapter, a generalized family of distribution is introduced by compounding Odd

BXII (Alizadeh et al.,2016) and Poission-G distribution. The cdf of odd Burr XII (OB) family

of distributions is.

Bc,k(x) =

FX (x,ξ)

1−FX (x,ξ)∫

0

c k xc−1 (1 + xc)−k−1 dx

= 1−{

1 +(

FX(x, ξ)1− FX(x, ξ)

)c}−k

. (4.1)

where FX(x, ξ) denotes the cdf of the baseline distribution. The pdf corresponding to Eq.

(4.1) is given by

bc,k(x) = c k fX(x)F c−1

X (x)

FXc+1(x)

{1 +

(FX(x, ξ)

1− FX(x, ξ)

)c}−k−1

,

where fX(x) = ∂FX(x)/∂x and FX(x) = 1− FX(x).

Gomes et al. (2015) recently introduced exponentiated-G Poisson(EGP) family of distribu-

tions. The cdf of EPG family is given by

F (x;λ, α) =1− exp [−λGα(x)]

1− e−λ,

50


where λ > 0, α > 0 and G(x) is the cdf of a random variable. Let α = 1, then the cdf and

pdf are, respectively, given by

F (x;λ, α) =1− exp [−λG(x)]

1− e−λ(4.2)

and

f(x; λ, α) = λg(x)exp [−λG(x)]

1− e−λ.

Here a compound family of distribution that is odd Burr G Poisson (OBGP), is proposed

by using the cdfs given in Eq. (4.1) and Eq. (4.2).

The physical interpretation of the proposed model is as follows. Suppose that a system has

N subsystems functioning individually at a given time, where N is a truncated Poisson

chance variable with probability mass function (pmf).

P (N = n) =λn

(eλ − 1)n!

for n = 1, 2, .... Let X represents the time of disaster of the first out of the N functioning

systems discribe by the independent random variable (Y1, ..., YN ) ∼ OB given by the cdf

(4.1). Then X = min(Y1, ..., YN ), so the conditional cdf of X (for x > 0) given N is

F (x|N) = 1− P (X > x|N) = 1− P (Y1 > x, ..., YN > x)

= 1− PN (Y1 > x) = 1− [1− P (Y1≤x)]N

= 1−[{

1 +(

FX(x, ξ)1− FX(x, ξ)

)c}−k]N

where c, k > 0. The unconditional cdf of X is

F (x) =e−λ

1− e−λ

∞∑

n=1

{1−

[{1 +

(FX(x, ξ)

1− FX(x, ξ)

)c}−k]n}

λn

n!

Using Eq. (4.1), we have

F (x) =1

1− e−λ

∞∑

n=1

{1− [1−Bc,k(x)]n} λn

n!

In more simplified form, the cdf of OBGP can be written as

F (x) =1− exp {−λBc,k(x)}

1− e−λ(4.3)

51


The pdf, Sf and hrf are given by

f(x) =λ bc,k(x)1− e−λ

exp {−λBc,k(x)} (4.4)

F (x) =exp {−λBc,k(x)} − e−λ

1− e−λ, (4.5)

and

h(x) =λ bc,k(x) exp {−λBc,k(x)}exp {−λBc,k(x)} − e−λ

.

The qf of OBGP family can be obtained by inverting Eq. (4.3)

QX(u) = F−1X

[(1 + z)−

1k − 1

] 1c

1 +[(1 + z)−

1k − 1

] 1c

, (4.6)

where z = − 1λ ln

{1− (1− e−λ)u

}and u ∼ Uniform(0, 1).

4.2 Special models of OBGP family

In this section, four special models of the OBGP family of the distributions are considered.

Their density and hazard rate functions plots are displayed to have a clue of the flexibility

of OBGP family density and hazard rate shapes. In the following models λ, c , k are the

parameters of the family.

4.2.1 Odd Burr-Weibull Poisson (OBWP) distribution

If Weibull distribution is the baseline distribution having cdf FX(x) = 1 − exp[−α xβ

],

with α > 0 and β > 0. Then the cdf and pdf of OBWP distribution are, respectively, given

by

F (x) =1− exp

{−λ

[1−

(1 +

[eαxβ − 1

]c)−k]}

1− e−λ, (4.7)

and

f(x) =λ c k α β xβ−1 e−αxβ

[1− e−αxβ

]c−1

(1− e−λ)[e−αxβ

]c+1 (1 +

[eαxβ − 1

]c)k+1

exp{−λ

[1−

(1 +

[eαxβ − 1

]c)−k]}

.

52


(i) If β = 1 in Eq. (4.7), then OBWP reduces to odd Burr Exponential Poisson (OBEP)

distribution, (ii) if c = 1 and k = 1 in Eq. (4.7), then OBWP reduces to Weibull poisson

distribution, (iii) if c = k = β = 1 in (4.7), then OBWP reduces to Exponential poisson (EP)

distribuiton. In Figure 4.1 the plots of density and hrf of OBWP distribution are dispa-

lyed. The possible shapes of the density of OBGP are left, right skewed, symmetrical and

reversed-J. The hrf shapes are increasing, decreasing, upside-down bathtub and bathtub.

(a) (b)

0.0 0.5 1.0 1.5 2.0 2.5

0.0

0.5

1.0

1.5

2.0

x

pdf

c = 2 k = 0.2 λ = 1.5 α = 2 β = 1.5c = 2 k = 0.2 λ = 0.2 α = 5 β = 0.5c = 0.8 k = 0.1 λ = 1.5 α = 0.5 β = 5c = 2 k = 0.2 λ = 2 α = 0.5 β = 4

0.0 0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

2.0

x

hrf

c = 1 k = 0.2 λ = 1.5 α = 2 β = 1.5c = 1.5 k = 0.2 λ = 0.2 α = 5 β = 0.5c = 0.08 k = 0.1 λ = 2 α = 0.8 β = 5c = 3 k = 0.07 λ = 1.5 α = 2 β = 1.5

Figure 4.1: Plots of (a) density and (b) hrf of OBWP distribution for some parameter values.

4.2.2 Odd Burr Lomax Poisson (OBLxP) distribution

If Lomax distribution is the base distribution with cdf FX(x) = 1−(1 + x

β

)−α, with α > 0

and β > 0. Then the cdf and pdf of BLxP distribution, respectively, are given by

F (x) =1− exp

{−λ

[1−

(1 +

[(1 + x

β

)α− 1

]c)−k]}

1− e−λ, (4.8)

and

f(x) =λ c k α

(1 + x

β

)α c−1[1−

(1 + x

β

)−α]c−1

(1− e−λ)[(

1 + xβ

)−α]c+1 (

1 +[(

1 + xβ

)α− 1

]c)k+1

exp

{−λ

[1−

(1 +

[(1 +

x

β

)α

− 1]c)−k

]}, (4.9)

(i) If c = 1 and k = 1 in Eq. (4.8), then OBLxP reduces to Lomax poisson (LxP) distribution,

(ii) if k = 1 in Eq. (4.8), then OBLxP reduces to log-logistic Lomax poisson (LLLxP) dis-

tribution. In Figure 4.2, the plots of density and hrf of OBLxP distribution are displayed.

53


The possible shapes of the density of OBLxP are right skewed and reversed-J. The hrf of

OBLxP are decreasing and upside-down bathtub.

(a) (b)

0.0 0.5 1.0 1.5 2.0 2.5

0.0

0.5

1.0

1.5

2.0

x

pdf

c = 2 k = 2 λ = 1.5 α = 2 β = 1.5c = 0.5 k = 1.5 λ = 0.5 α = 2 β = 2c = 3 k = 0.4 λ = 5 α = 3 β = 1.5c = 6 k = 0.2 λ = 3 α = 3 β = 1.7

0 1 2 3 4 5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

x

hrf

c = 3 k = 0.4 λ = 5 α = 3 β = 1.5c = 0.5 k = 1.5 λ = 0.5 α = 2 β = 2c = 4 k = 0.3 λ = 3 α = 2 β = 2c = 2 k = 0.2 λ = 0.2 α = 0.2 β = 0.7

Figure 4.2: Plots of (a) density and (b) hrf of OBLxP distribution for some parameter values.

4.2.3 Odd Burr gamma Poisson distribution (OBGaP)

If gamma distribution as the base distribution having cdf FX(x) =γ�α, x

β

�

Γ(α) = P(α, x

β

),

with α > 0 and β > 0. Then the cdf and pdf of OBGaP distribution, respectively, are given

by

F (x) =

1− exp

−λ

1−

(1 +

[P�α, x

β

�

1−P�α, x

β

�]c)−k

1− e−λ, (4.10)

and

f(x) =λ c k βα xα−1 e−β x

[P

(α, x

β

)]c−1

Γ(α) (1− e−λ)[1− P

(α, x

β

)]c+1(

1 +

[P�α, x

β

�

1−P�α, x

β

�

]c)k+1

exp

−λ

1−

1 +

P

(α, x

β

)

1− P(α, x

β

)

c−k

.

(i) If c = 1 in Eq. (4.10), then OBGaP reduces to Odd Lomax gamma poisson (OLxGaP)

distribution, (ii) if k = 1 in Eq. (4.10), then OBGaP reduces to Odd Log-logistic gamma

poisson (OLLGaP) distribution and (iii) if c = k = 1 in Eq. (4.10), then OBGaP reduces to

54


Exponential poisson (EP) distribution. In Figure 4.3, the plots of density and hrf of OBGaP

distribution are dispalyed. The possible shapes of the density of OBGaP are right skewed,

symmetrical and reversed-J. The hrf of OBBP are decreasing, upside-down bathtub and

bathtub.

(a) (b)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

x

pdf

c = 1.5 k = 0.5 λ = 4 α = 3 β = 1.5c = 0.5 k = 0.3 λ = 3 α = 2 β = 2c = 3 k = 0.1 λ = 0.2 α = 1.5 β = 2c = 2 k = 5 λ = 8 α = 2 β = 0.3

0 1 2 3 4 5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

x

hrf

c = 1.5 k = 0.5 λ = 4 α = 3 β = 1.5c = 0.3 k = 5 λ = 0.5 α = 1.8 β = 2c = 0.5 k = 3 λ = 3 α = 2 β = 2c = 0.1 k = 5 λ = 0.5 α = 2 β = 0.3c = 1.5 k = 0.5 λ = 4 α = 5 β = 2

Figure 4.3: Plots of (a) density and (b) hrf of OBGaP distribution for some parameter val-

ues.

4.2.4 Odd Burr beta Poisson (OBBP) distribution

If beta distribution as the baseline distribution having cdf FX(x) = BX(α,β)B(α,β) = IX (α, β),

with α > 0 and β > 0. Then the cdf and pdf of OBBP distribution, respectively, are given

by

F (x) =1− exp

{−λ

[1−

(1 +

[IX(α,β)

1−IX(α,β)

]c)−k]}

1− e−λ, (4.11)

and

f(x) =λ c k [IX (α, β)]c−1

B (α, β) (1− e−λ) [1− IX (α, β)]c+1(1 +

[IX(α,β)

1−IX(α,β)

]c)k+1

exp

{−λ

[1−

(1 +

[IX (α, β)

1− IX (α, β)

]c)−k]}

.

(i) If c = 1 in (4.11), then OBBP reduces to Odd Lomax beta poisson (OLxBP) distribution,

(ii) if k = 1 in (4.11), then OBBP reduces to Odd Log-logistic beta poisson (OLLBP) distri-

bution and (iii) if c = k = 1 in (4.11), then OBBP reduces to beta poisson (BP) distribution.

55


In Figure 4.4, the plots of density and hrf of OBBP distribution are displayed. The possible

shapes of the density of OBBP are right, left skewed, symmetrical and U-shapes. The hrf

of OBBP are increasing, decreasing and bathtub.

(a) (b)

0.0 0.2 0.4 0.6 0.8 1.0

01

23

4

x

pdf

c = 2 k = 3 λ = 3 α = 1.5 β = 0.7c = 0.2 k = 0.5 λ = 2 α = 2 β = 3c = 2 k = 3 λ = 3 α = 1.5 β = 0.3c = 1.5 k = 2 λ = 0.5 α = 1 β = 2

0.0 0.2 0.4 0.6 0.8

0.0

0.5

1.0

1.5

2.0

2.5

x

hrf

c = 1.5 k = 0.5 λ = 4 α = 3 β = 1.5c = 0.3 k = 5 λ = 0.5 α = 1.8 β = 2c = 0.5 k = 3 λ = 3 α = 2 β = 2c = 0.1 k = 5 λ = 0.5 α = 2 β = 0.3c = 1.5 k = 0.5 λ = 4 α = 5 β = 2

Figure 4.4: Plots of (a) density and (b) hrf of OBBP distribution for some parameter values

4.3 Some mathematical properties of OBGP family

Here, some useful mathematical properties of OBGP family of distribution are studied.

4.3.1 Infinite mixture representation

Here, a infinite mixture representation of cdf and pdf of the OBGP family is obtained in

terms of the baseline cdf and pdf.

Theorem 4.3.1. If X ∼ OBGP(λ, c, k ξ), we have the following approximation.

For λ, c, k > 0 be the real non-integer values, then we have the following mixture representation.

F (x) =∞∑

q=0

aq Hq(x), (4.12)

where Hq(x) = F qX(x; ξ) represents the exp-G distribution with power parameter q, and the coeffi-

56


cients are defined by

aq =∞∑

i=1

∞∑

j,l,m=0

(−1)i+j+1 λi

i!(1− e−λ)

i

j

kj + l − 1

l

cl + m− 1

m

Sq(m + cl)

Sq(m + cl) =∞∑

r=q

m + cl

r

r

q

(−1)r+q (4.13)

Eq.(4.12) reveals that the OBGP distribution can be expressed as the infinite mixture combination

of the base line pdf and cdf.

For λ, c, k > 0 be the real non-integer, we have

f(x) =∞∑

q=0

aq+1hq+1(x), (4.14)

where aq+1 are defined in (4.13).

proof:

If b > 0 is real number, then we have generalized binomial theorem

(1− z)−k =∞∑

i=0

k + i− 1

i

zi. (4.15)

and Taylor series expansion as

1− e−x =∞∑

i=1

(−1)i+1 xi

i!(4.16)

Using series expansion in Eq.(4.16), we obtain

F (x) =1

1− e−λ

∞∑

i=1

(−1)i+1λi

i!Bi

c,k(x)

Consider

Bic,k(x) =

[1−

{1 +

(FX(x)

1− FX(x)

)c}−k]i

Using series expansion in Eq.(4.15) we have

Bic,k(x) =

i∑

j=0

i

j

(−1)j

∞∑

l=0

k j + l − 1

l

(−1)l

∞∑

m=0

c l + m− 1

m

Fm+c l

X (x)

57


Again consider

Rm+c l(x) = [1− {1− FX(x)}]m+c l

Using series expansion in Eq. (4.15) we have

Rm+c l(x) =∞∑

r=0

r∑

q=0

m + c l

r

r

q

(−1)r+q F q

X(x)

=∞∑

q=0

∞∑r=q

m + c l

r

r

q

(−1)r+q F q

X(x)

Now Eq. (4.3) becomes

F (x) =∞∑

q=0

aq Hq(x)

where aq is given in (4.13) and Hq(x) = F qX(x; ξ) is the exp-G density function with ξ

parametric space. Eq. (4.14) can easily be obtained by simple derivative of Eq. (4.12) .

4.3.2 Shapes

The shapes of the density and hrf can be described analytically. The critical points of the

OBGP density function are the roots of the equation:

r′(x)fX(x)

− (c− 1)fX(x)

1− FX(x)− (c + 1)

fX(x)FX(x)

− (k + 1)z′izi− λ

[k (1 + zi)−k−1 z′i

]= 0

above equation may have more than one root.

The critical point of hrf are obtained from equation:

r′(x)fX(x)

+ (c− 1)fX(x)FX(x)

+ (c + 1)fX(x)

1− FX(x)− (k + 1)

z′i1 + zi

− λ[k (1 + zi)−k−1 z′i

]

+

exp

[−λ{1− (1 + zi)

−k}]λk (1 + zi)

−k−1 z′i

exp[−λ{1− (1 + zi)

−k}]− e−λ

,

where zi =(

FX(xi)1−FX(xi)

)and z′i = d

dx

(FX(xi)

1−FX(xi)

).

58


4.3.3 Moments

The rth moment of the OBGP family of distributions can be obtained by following expres-

sion

E(Xr) =∞∑

q=0

aq+1

∞∫

0

xr hq+1(x)dx (4.17)

where aq+1 are defined in (4.13).

The mth incomplete moment of the OBGP family of distributions can be obtained as

µm(x) =∞∑

q=0

aq+1T′m(x), (4.18)

where T ′m(x) =x∫0

xr hq+1(x)dx.

The moment generating function of the OBGP family of distributions can be defined by

following expression as

MX(t) =∞∑

q=0

aq+1Mq+1(t), (4.19)

where Mq+1(t) =∞∫0

et x hq+1(x)dx.

The mean deviations of the OBGP family of distributions about the mean and median are,

respectively, defined as

Dµ = 2µF (µ)− 2µ1(µ) (4.20)

DM = µ− 2µ1(M) (4.21)

where µ = E(X) can be obtained form Eq. (2.14), M = Median(X) can be obtained

form Eq. (4.6), F (µ) can be calculated easily from Eq. (4.3) and µ1(.) can be obtained from

Eq. (4.18).

4.4 Entropies

Here, two entropies, Renyi and shannon are considered.

59


Theorem 4.4.1. If X ∼ OBGP(λ, c, k), then we have following approximation.

A. For δ > 0 and λ, c, k > 0 be the real non-integer values. Then we have following expression

for Rayni entropy.

IR =1

1− δ

log K + log

∞∑

m=0

Vm,c(δ, l)

∞∫

0

f δX(x) F

m+c (l+δ)−δX (x) dx

,

where rδ(x) represents the pdf of the baseline distribution with power parameter δ and Rm+c (l+δ)−δ(x)

is the cdf of the baseline distribution with power parameter m + c (l + δ) − δ. The above integral

only depends on the baseline cdf and pdf . The coefficients are defined as

Vm,c(δ, l) =∞∑

i,m=0

i∑

j=0

i

j

k(δ + j) + δ + l − 1

m

c l + δ (c + 1) + m− 1

m

(−1)i+j+l (λ δ)i

i!

B. For g(x) be the density of the OBGP family of distributions. We have shannon entropy of

OBGP family as

ηx = M +[1− e−λ

]− E [log fX(x)]− (c− 1)E [log FX(x)] + (c− 1)E

[log FX(xi)

]

+ (k + 1)∞∑

i=0,j=1

ai,j(c) E[F c j+i

X (x)]

+ λ

∞∑

i=0,j=1

bi,j(c, k) E[F c j+i

X (x)],

where r(x), R(x) represents the pdf and cdf of the base line densities. The above expectations only

depends on the baseline densities. The coefficients are defined as

ai,j(c) =(−1)j+1

j

c + i− 1

i

(4.22)

bi,j(c, k) = −(−1)i+1

j

k + j − 1

j

c j + i− 1

i

(4.23)

Proof of A: The Renyi entropy of OBGP family of distribution is given by

IR =1

1− δlog

∞∫

0

[λ bc,k(x)1− e−λ

exp {−λBc,k(x)}]δ

dx (4.24)

Using series expansion in Eq. (4.16), we obtain

exp [−λ δ Bc,k(x)] =∞∑

i=0

(−1)i (λ δ)i

i!Bi

c,k(x) (4.25)

60


Using series expansion in Eq. (4.15), we have

exp [−λ δ Bc,k(x)] =∞∑

i=0

(−1)i (λ δ)i

i!

[1−

{1 +

(FX(x)

1− FX(x)

)c}−k]i

=∞∑

i=0

(−1)i (λ δ)i

i!

i∑

j=0

i

j

(−1)j

×{

1 +(

FX(x)1− FX(x)

)c}−k j

(4.26)

Combining the result in Eq. (4.26), we have

f δ(x) = (c k)δ∞∑

i=0

(−1)i (λ δ)i

i!

i∑

j=0

i

j

(−1)j f δ

X(x)F

δ(c−1)X (x)

Fδ(c+1)X (x)

×{

1 +(

FX(x)1− FX(x)

)c}−k(δ+j)−δ

Using series expansion in Eq. (4.15) we have

f δ(x) = (c k)δ∞∑

i,l=0

∞∑

m=0

i

j

k(δ + j) + δ + l − 1

l

× cl + δ(c + 1) + m− 1

m

(−1)i+j+l (λ δ)i

i!f δ

X(x) Fm+c(l+δ)−δX (x)

where f δX(x) and F

m+c(l+δ)−δX (x) represents the the base line pdf and cdf, with power pa-

rameter δ and m + c(l + δ)− δ respectively.

Proof of B: We have following Taylor series expansions

log(1 + x) =∞∑

j=1

(−1)j+1

jxj (4.27)

The shannon entropy of OBGP family of distribution is

ηx = − log(λ c, k) + log(1− eλ)−E(log fX(xi))− (c− 1)E(log FX(xi)) + (c− 1)E(log FX(xi))

+ (k + 1)E

[log

{1 +

(FX(x)

1− FX(x)

)c}]+ λE

[1−

{1 +

(FX(x)

1− FX(x)

)c}−k]

(4.28)

Using Eq. (4.16), we have

log{

1 +(

FX(x)1− FX(x)

)c}=

∞∑

j=1

(−1)j+1

j

[FX(x)

1− FX(x)

]c j

61


Now using expansion in Eq. (4.15), we obtain

log{

1 +(

FX(x)1− FX(x)

)c}=

∞∑

j=1

(−1)j+1

j

∞∑

i=0

c j + i− 1

i

F c i+l

X (x)

Now using expansion in Eq. (4.15), we obtain

1−{

1 +(

FX(x)1− FX(x)

)c}−k

= −∞∑

j=1

k + j − 1

j

(−1)j

[FX(x)

1− FX(x)

]c j

Again using expansion in Eq. (4.15), we obtain

= −∞∑

j=1

k + j − 1

j

(−1)j

∞∑

i=0

c j + i− 1

i

F c i+l

X (x)

Eq. (4.28) can be written as

ηx = M +[1− e−λ

]− E [log fX(x)]− (c− 1)E [log FX(x)] + (c− 1)E

[log FX(xi)

]

+ (k + 1)∞∑

i=0,j=1

ai,j(c) E[F c j+i

X (x)]

+ λ∞∑

i=0,j=1

bi,j(c, k) E[F c j+i

X (x)]

The coefficients are defined in Eq.(4.23) and Eq. (4.23).


If X1 ∼ OBGP(c, k, β, λ1) and X2 ∼ OBGP(c, k, β, λ2), then

f(x) =λ1 bc,k(x)1− e−λ1

exp {−λ1Bc,k(x)} (4.29)

and

g(x) =λ2 bc,k(x)1− e−λ2

exp {−λ2Bc,k(x)} (4.30)

Then their ratio[

f(x)g(x)

]will be

f(x)g(x)

=λ1

λ2

1− e−λ2

1− e−λ1exp {−(λ1 − λ2)Bc,k(x)}

Taking derivative with respecto to x, we have

d

dx

f(x)g(x)

=λ1

λ2(λ1 − λ2)

1− e−λ2

1− e−λ1exp {−(λ1 − λ2)Bc,k(x)} B′

c,k(x),

where B′c,k(x) = c k r(x) Rc−1(x)

Rc+1(x)

{1 +

(R(x,ξ)

1−R(x,ξ)

)c}−k−1. From the above equation we ob-

serve that, if λ1 < λ2 ⇒ ddx

f(x)g(x) < 0, then this implies that likelihood ratio exists between

X ≤lr Y .

62



Here, the expression of the ith order statistics is defined as a infinite series of baseline

densities.

Theorem 4.5.1. If n is an integer value and for i = 1, 2, ..., n and X1, X2, ..., Xn be identi-

cally independently distributed random variables. Then the density of ith order statistics of OBGP

distribution is

fi:n(x) =n−i∑

j=0

∞∑

q,t=0

mj,q,t hq+t(x), (4.31)

where

mj,q,t =n!(−1)j aq+1 dt:j+i−1

(i− 1)!(n− i− j)!j!(q + t + 1(4.32)

and hm(x) = [m + 1] r(x)Rm(x).

Proof:

If n ≥ 1 is an integer value then, we have following power series expansion (Gradshtegn

and Ryzhik, 2000).( ∞∑

k=0

ak xk

)n

=∞∑

k=0

ak:n xk, (4.33)


m a0

m∑k=1


The expression for ith order statistics is defined as

fi:n(x) =n!

(i− 1)!(n− i)!g(x) Gi−1(x) [1−G(x)]n−i

Using the series expansion in Eq. (3.17), we get

fi:n(x) =n!

(i− 1)!(n− i)!

n−i∑

j=0

n− i

j

(−1)if(x)[F (x)]i+j−1.

Using the infinite mixture representation of OBGP densities in Eqs. (4.12), (4.14) and (4.33),

we get

fi:n(x) =n−i∑

j=0

∞∑

q,t=0

mj,q,t hq+t(x),

where mj,q,t are defined in Eq. (4.32).

63


4.6 Maximum Likelihood method

If x1, x2, ..., xn be a random sample of size n from the OBGP family given in Eq. (4.30)

distribution, then the log-likelihood function for the vector of parameter Θ = (c, k, β, s, ξ)T

is

l(Θ) = n log(λ c k)− n log(1− e−λ) +n∑

i=1

log r(xi) + (c− 1)n∑

i=1

log R(xi)

−(c + 1)n∑

i=1

log R(xi)− (k + 1)n∑

i=1

log zi − λn∑

i=1

{1− z−ki }, (4.34)

where zi ={

1 +(

1−R(x,ξ)R(x,ξ)

)c}.


Uλ =n

λ+

[n e−λ

1− e−λ

]−

n∑

i=1

{1− z−ki },

Uk =n

k−

n∑

i=1

log zi − λ kn∑

i=1

z−k−1i z′i,

Uc =n

c+

n∑

i=1

log R(xi)−n∑

i=1

log R(xi)− (k + 1)n∑

i=1

[z′i:czi

]− λ k

n∑

i=1

z−k−1i z′i:c,

Uξ =n∑

i=1

[rξ(xi)r(xi)

]+ (c− 1)

n∑

i=1

[Rξ(xi)R(xi)

]+ (c− 1)

n∑

i=1

[Rξ(xi)

1−R(xi)

]− (k + 1)

n∑

i=1

[z′i:ξzi

]

− λ kn∑

i=1

z−k−1i z′i:ξ.

Setting Uλ, Uc, Uk and Uξ equal to zero and solving these equations simultaneously yields


4.7 Properties of OBLP distribution

Here, properties of special model Odd Burr Lomax Poisson distribution are given in detail.

Mixture representation of OBLxP can be obtained from Eqs. (4.12) and (4.14).

F (x) =∞∑

q=0

aq

{1−

(1 +

x

β

)−α}q

.

f(x) =∞∑

m=0

aq+1(q + 1)α

β

(1 +

x

β

)−α−1{

1−(

1 +x

β

)−α}q

.

64


The qf of OBLxP distribution can be obtained from Eq. (4.6)

Qx(u) = β

{(1 + z)−

1k − 1

} 1c

1 +{

(1 + z)−1k − 1

} 1c

− 1α

− 1

,

where z = − 1λ ln

{1− (1− e−λ)u

}.

The rth moment of OBLxP distribution can be obtained from Eq. (4.17)

µ′r =∞∑

m=0

aq+1 (q + 1)q∑

s=0

q

s

(−1)s α βrB (α(s + 1)− r; r + 1)

The mth moment of OBLxP distribution can be obtained from Eq. (4.18)

µm =∞∑

m=0

aq+1 (q + 1)q∑

s=0

q

s

(−1)s α βmB x

β(α(s + 1)−m;m + 1)

wherex∫0

xa−1(1− x)b−1 = Bx (a, b) is the incomplete beta function.

The mgf of OBLxP distribution can be obtained from Eq. (4.19)

M0(t) =∞∑

m=0

aq+1 (q + 1)q∑

s=0

q

s

(−1)s e−t Γ(−α(s + 1)) (−tβ)α(s+1)

First incomplete moment of OBLxP distribution can be obtained, by setting m = 1 in above

equation, we get the

µ1 =∞∑

m=0

aq+1 (q + 1)q∑

s=0

q

s

(−1)s α β1B x

β(α(s + 1)− 2; 2)

Above expression can be used to obtain mean deviation about mean and median, respec-

tively, from Eqs. (4.20) and (4.21).

Let x1, ..., xn be a sample of size n from the OBLxP distribution, then the log-likelihood

function for the vector of parameters Θ = (λ, c, k, α, β) is

l(Θ) = n log(λ c k α)− n log(1− e−λ

)+ (cα− 1)

n∑

i=1

log(

1 +xi

β

)+ (c− 1)

×n∑

i=1

log

{1−

(1 +

xi

β

)−α}− (k + 1)

n∑

i=1

log[1 +

{(1 +

xi

β

)α

− 1}c]

− λ

×n∑

i=1

{1−

[1 +

{(1 +

xi

β

)α

− 1}c]−k

},

65


where zi ={

(1 + xiβ )α − 1

}c.


Uλ =n

λ+

n e−λ

1− e−λ−

n∑

i=1

{1− [1 + zi]

−k}

Uk =n

k−

n∑

i=1

log(1 + zi)− λn∑

i=1

[{1 + zi}−k log {1 + zi}

]

Uc =n

c+ b

n∑

i=1

log(

1 +xi

β

)+

n∑

i=1

log

{1−

(1 +

xi

β

)−α}− (k + 1)

n∑

i=1

[z′i;c

1 + zi

]

− λn∑

i=1

[k (1 + zi)−k−1 z′i;c

]

Uα =n

α+ c

n∑

i=1

log(

1 +xi

β

)+ (c− 1)

n∑

i=1

(1 + xi

β

)−αlog

(1 + xi

β

)

1−(1 + xi

β

)−α

− (k + 1)n∑

i=1

[z′i;α

1 + zi

]+ k λ

n∑

i=1

[(1 + zi)−k−1 z′i;α

]

Uβ = −n

β− (c α− 1)

n∑

i=1

[xiβ2

1 + xiβ

]− (c− 1)

n∑

i=1

α(1 + xi

β

)α−1 [xiβ2

](1 + xi

β

)α

+ (k + 1)n∑

i=1

c{(

1 + xiβ

)α− 1

}c−1α

(1 + xi

β

)α−1 [xiβ2

]

1 +{(

1 + xiβ

)α− 1

}c

− λ

n∑

i=1

[z−k−1 z′i:β

]

The density of ith order statistics of OBLxP distribution can be obtained from Eq. (4.31)

fi:n(x) =n−i∑

j=0

∞∑

q,δ=0

mj,q,δ (q + δ + 1)α

β

(1 +

xi

β

)−α−1[1−

(1 +

xi

β

)−α]q+δ

4.7.1 Simulations study

The mean, variance and the mean squared error (MSE)of the maximum likelihood es-

timate were calculated for simulated samples. Various simulation studies for different

sample sizes n and combination of parameter values, generating 1000 random samples are

simulated. The observations denoted by x1, ..., xn were generated from the OBLxP dis-

tribution, where they were generated from the inverse transformation method. From the

66


simulation results, the data of which are shown in Tables 4.1 and 4.3, it was observed MSE

decreased when n increased. In relation to the relative bias, their values remained close in

all the scenarios. The greatest impact of the bias occurred with the parameters c, except,

when c assumes small values. Also, higher values of the bias occurred in the situation that

the size of n was smaller than 100 independent of the combinations.

Table 4.1: Mean, bias and MSEs of the estimates of the parameters of OBLxP for c = 10,

k = 0.06, λ = 4 and α = 9.n Parameters Mean Bias M.S.E

c 20.971 10.971 644.003

20 k 0.15392 0.12553 2.4452

λ 7.179 3.2156 64.242

α 8.959 -0.04056 1.20870

c 13.311 3.311 92.285

50 k 0.0627 0.0078 0.003

λ 5.629 1.6392 35.6023

α 9.071 0.0715 0.3839

c 11.207 1.2070 15.4224

100 k 0.06496 0.0085 0.0019

λ 4.994 1.0014 21.1794

α 9.055 0.0550 0.19278

c 10.778 0.7778 6.3137

150 k 0.0674 0.0106 0.0017

λ 4.400 0.4049 11.6308

α 9.046 0.0457 0.1267

67


Table 4.2: Mean, bias and MSEs of the estimates of the parameters of OBLxP model for

c = 10, k = 0.5, λ = 4 and α = 9.n Parameters Mean Bias M.S.E

c 15.656 5.656 472.02

20 k 1.8799 2.2987 27.0474

λ 6.93885 2.9426 154.568

α 8.705 -0.2955 3.2408

c 11.154 1.1544 11.189

50 k 1.0483 0.6174 4.0659

λ 5.629 0.2113 56.465

α 4.2110 0.06882 1.05690

c 10.649 0.6488 3.7530

100 k 0.7346 0.24177 0.69969

λ 3.77812 -0.2217 26.5329

α 9.147 0.1468 0.413603

c 10.395 0.3946 2.1428

150 k 0.6522 0.15219 0.127139

λ 3.56021 -0.4398 17.4689

α 9.140 0.1396 0.24970

Table 4.3: Mean, bias and MSEs of the estimates of the parameters of OBLxP for c = 0.5,

k = 0.06, λ = 4 and α = 9.n Parameters Mean Bias M.S.E

c 1.1445 0.64872 7.5180

20 k 0.19712 0.15151 0.26746

λ 8.748 4.76926 106.197

α 10.452 1.480 118.394

c 0.5465 0.04896 0.060262

50 k 0.15914 0.10855 0.181642

λ 7.142 3.1464 59.5708

α 10.288 1.3108 102.652

c 0.5259 0.0259 0.0254

100 k 0.1047 0.0495 0.0750

λ 5.906 1.9153 31.5131

α 10.092 1.121 69.568

c 0.5110 0.0119 0.0103

150 k 0.0808 0.0248 0.0181

λ 5.047 1.0558 17.1978

α 9.833 0.8370 45.604

68


Table 4.4: Mean, bias and MSE (Mean Square Error) of the estimates of the parameters of

OBLxP with c = 10, k = 0.06, λ = 0.5 and α = 9.n Parameters Mean Bias M.S.E

c 30.965 20.9696 1040.024

20 k 0.1011 0.0456 1.7718

λ 0.6295 0.2768 1.6450

α 8.807 -0.1934 1.8777

c 21.149 3.0542 11.154

50 k 0.0506 0.0080 -0.0068

λ 0.7142 0.3508 0.2726

α 8.956 0.3837 -0.0443

c 14.239 4.2389 125.676

100 k 0.05301 -0.0043 0.0009

λ 0.6580 0.1880 1.0192

α 8.984 -0.0160 0.4029

c 12.433 2.4326 58.295

150 k 0.0535 -0.003544 0.0006

λ 0.6087 0.1382 0.7445

α 9.003 0.0028 0.2833

4.8 Application

The applications on real data set is performed to explain the importance of the OBLxP

family of distribution. The model parameters are estimated by the ML method and three

goodness-of-fit statistics are calculated to compare the OBLxP distribution with Kw-Weibull

Poisson (Kw-WP)(Ramos, 2015),Beta Lomax (B-Lx) , Kumaraswamy Lomax (K-Lx) and Lo-

max distributions. The computations were performed using the package Adequacy Model

in R developed.

4.8.1 Data set 5: Failure times mechanical components

The data set is taken from the book ”Weibull models, series in probability and statistics”

by Murthy DNP et al.(2004). The corresponding data are referring to the failure times of

20 mechanical components.

69


Table 4.5: MLEs and their standard errors for data set 5.

Distribution c k λ β α

OBLxP 9.8829 0.0658 3.9548 6.1425 0.6561

(5.2939) (0.1119) (5.1296) (32.9497) (3.6866)

Kw-WP 1.3435 25.8359 5.1352 19.6074 0.1512

(0.0151) (0.1612) (2.0805) (6.3839) (0.0667)

B-Lx 67.5047 0.8771 0.1044 6.8834 -

(58.3961) (0.7190) (0.1250) (7.2679) -

K-Lx 47.5001 1.2606 0.0755 4.6266 -

39.4774 0.9579 0.0783 3.7908 -

Lx 5.4148 45.2542 - - -

(11.2841) (93.2701) - - -

Table 4.6: Model adequacy measures A∗ and W∗ for data set 5.

Distribution W* A*

OBLxP 0.0430 0.2846

Kw-WP 0.0638 0.4918

B-Lx 0.0769 0.5981

K-Lx 0.0851 0.6562

Lx 0.2818 1.8519

(a) (b)

x

Den

sity

0.1 0.2 0.3 0.4 0.5

05

1015

20 OBLxPKw−WPB−LxKw−LxLx

0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

1.0

x

cdf

OBLxPKw−WPB−LxKw−LxLx

Figure 4.5: Plots of estimated pdf and cdf of OBLxP distribution

70


4.9 Conclusions and Results

In this chapter, a family of distributions called ”Odd Burr XII G poisson family of dis-

tributions” is proposed. Most of its mathematical properties such as, rth moment, sth

incomplete moment, moment generating function, mean deviations, stochastic ordering,

Rayni and Shannon entropies, order statistics and estimation of parameters by ML method

are carried out. A special model is discussed in detail. An application is carried out on real

data set to check the performance of the proposed family, which provides consistently bet-

ter fit than other competitive models.

71

Chapter 5

A New Generalized Burr Distribution

based on quantile function

5.1 Introduction

The revolutionary idea on parameter induction was introduced by Alzaatreh et al. (2013)

by defining Transformed-Transformer (T-X) family of distributions. Let r(t) be the prob-

ability density function (pdf) of a random variable T ∈ [a, b] for −∞ ≤ a < b < ∞ and

let F (x) be the cdf of a random variable X such that the transformation W (·) : [0, 1] −→[a, b] satisfies the following conditions: (i) W (·) is differentiable and monotonically non-

declining, and(ii) W (0) → a and W (1) → b.

Alzaatreh et al. (2013) proposed the cdf of the T-X family of distributions by

G(x) =∫ W [F (x)]

ar(t) dt. (5.1)

If T ∈ (0,∞), X is a continuous random variable and W [F (x)] = − log[1 − F (x)]. Then,

the pdf corresponding to Eq. (5.1) is given by

g(x) =f(x)

1− F (x)r(− log

[1− F (x)

])= hf (x) r

(Hf (x)

), (5.2)

where hf (x) = f(x)1−F (x) and Hf (x) = − log[1 − F (x)] are the hrf and chrf corresponding to

any baseline pdf f(x), respectively.

Let T , R and Y be three random variables with their cdf FT (x) = P (T ≤ x), FR(x) =

72


P (R ≤ x) and FY (x) = P (Y ≤ x). The quantile function of these three cdf’s are QT (u),

QR(u) and QY (u), where quantile function is defined as QZ(u) = inf{z : FZ(z) ≥ u}, 0 <

u < 1. The densities of T , R and Y are denoted by fT (x), fR(x) and fY (x), respectively.

We assume the random variables T ∈ (a, b) and Y ∈ (c, d), for −∞ ≤ a < b ≤ ∞ and

−∞ ≤ c < d ≤ ∞. Aljarrah et al. (2014) (See also Alzaatreh et al. (2014)) defined the cdf of

the T-R{Y} family by

FX(x) =∫ QY (FR(x))

afT (t) dt = FT {QY (FR(x))} . (5.3)

The pdf and hrf corresponding to Eq. (5.3), are given by

fX(x) = fR(x)fT {QY [FR(x)]}fY {QY [FR(x)]} .

or

fX(x) = fR(x) Q′Y (FR(x)) fT {QY [FR(x)]}

and

hX(x) = hR(x)× hT {QY [FR(x)]}hY {QY [FR(x)]} .

If a random variable R follows the BXII distribution, then cdf and pdf of T-Burr{Y} family


FX(x) =

QY (1−(1+xc)−k)∫

a

fT (t)dt = FT (QY (1− (1 + xc)−k)). (5.4)

and

fX(x) = c k xc−1 (1 + xc)−k−1fT

(QY (1− (1 + xc)−k)

)

fY

(QY (1− (1 + xc)−k)

) , (5.5)

Table 5.1 contains qfs. of well known distributions. Different generalized Burr families of

T-Burr{Y} can be generated by using these qfs.

Remark 1. If X follows the T-Burr{Y} family of distributions given in Eq. (5.4), then we have the

followings:

73


Table 5.1: qfs. for different distributions.

S.No Y QY (u)

1. Lomax β[(1− u)−

1α − 1

]

2. Weibull[−α−1 ln(1− u)

] 1β

3. Log-Logistic α[u−1 − 1

]− 1β

(i) Xd=

{[1− FY (T )

]− 1k − 1

}1c

(ii) QX(u) ={[

1− FY

(QT (u)

)]− 1k − 1

}1c

(iii) if Td= Y , Then X

d= Burr(c, k), and

(iv) if Yd= Burr(c, k), then X

d= T .

5.1.1 T-Burr{Lomax} Family of distributions

Using qf. of Lomax distribution in Table 5.1, the cdf and pdf of T-Burr{Lomax} are, re-

spectively, given by

FX(x) = FT

{β

{(1 + xc)

kα − 1

}}, (5.6)

If α = 1 in above equation, then

FX(x) = FT

{β

{(1 + xc)k − 1

}}. (5.7)

and

fX(x) = β burr(c,−k) fT

{β

{(1 + xc)k − 1

}},

where burr(c,−k) = c k xc−1 (1+xc)k−1.

5.1.2 T-Burr{log-Logistic} Family of distributions

Using qf. of log-logistic distribution in Table 5.1, the cdf and pdf of T-Burr{log-Logistic}are, respectively, given by

FX(x) = FT

{α

[(1 + xc)k − 1

] 1β

}. (5.8)

74


and

fX(x) =α

βburr(c, k)

[(1 + xc)k − 1

] 1β−1

fT

{α

[(1 + xc)k − 1

] 1β

}.

5.1.3 T-Burr{Weibull} Family of distributions

Using qf. of Weibull distribution in Table 5.1, the cdf and pdf of T-Burr{Weibull} are,

respectively, given by

FX(x) = FT

{[k

αln (1 + xc)

] 1β

}, (5.9)

If α = 1 in above equation, then

FX(x) = FT

{[k ln (1 + xc)]

1β

}. (5.10)

and

fX(x) =c k xc−1

β (1 + xc)[k ln (1 + xc)]

1β−1

fT

{[k ln (1 + xc)]

1β

}. (5.11)

5.2 Some properties of the T-Burr{Y} family of distributions

Here, some statistical properties of T-Burr {Y} family of distributions mode(s), rth mo-

ments, Shannon entropy and mean deviations are studied.

5.2.1 Mode

Theorem 5.2.1. For f(x) be the pdf of T -Burr{Y } family of distributions, then for d2

d x2 f(x) <

0, we have

x = (c−1){

c (k + 1)xc−1

1 + xc−Ψ[Q′

Y {1− (1 + xc)−k}]−Ψ{fT [QY {1− (1 + xc)−k}]}}

(5.12)

where Ψ(f) = f ′f .

Proof:

Consider

fX(x) = c k cc−1 (1 + xc)−k−1 Q′Y [1− (1 + xc)−k] fT {QY [1− (1 + xc)−k].}

75


Taking log on both sides, we obtain

log fX(x) = log[c k cc−1 (1 + xc)−k−1

]+log Q′

Y [1−(1+xc)−k]+log fT {QY [1−(1+xc)−k].}

Taking derivative with respect to x, we obtain

d

d xlog fX(x) = log(c k)+(c−1) log x−(k+1) log(1+xc)+

Q′′Y [1− (1 + xc)−k]

Q′Y [1− (1 + xc)−k]

+f ′T {QY [1− (1 + xc)−k]}fT {QY [1− (1 + xc)−k]} .

Setting dd x log fX(x) = 0, we obtain

(c−1) log x−(k+1) log(1+xc)+Ψ{

Q′Y [1− (1 + xc)−k]

}+Ψ

{fT

[QY {1− (1 + xc)−k}

]}= 0.

The above equation can be written as

x = (c− 1){

c (k + 1)xc−1

1 + xc−Ψ[Q′

Y {1− (1 + xc)−k}]−Ψ{fT [QY {1− (1 + xc)−k}]}}

.

5.2.2 Moments

The rth moment of T-Burr{Y} can be obtained using Remark 1(i)

E(Xr) = E[{1− FY (T )}− 1

k − 1] r

c.

Using generalized binomial theorem, (x + y)r =∞∑

j=0

r

j

xr−j yj (|x| > |y|), we obtain

E(Xr) =∞∑

j=0

rc

j

(−1)j E {1− FY (T )}− (r−j)

k . (5.13)

Using the expression in Eq. (5.13) the rth moment of T-Burr{Lomax}, T-Burr{Log-logistic}and T-Burr{Weibull} distributions can be obtain, respectively, as

E(Xr) =∞∑

j=0

rc

j

(−1)j E

{(1 +

T

β

) 1k(r−j)

}, (5.14)

E(Xr) =∞∑

j=0

rc

j

(−1)j E

(1 +

(T

β

)β) 1

k(r−j)

, (5.15)

E(Xr) =∞∑

j=0

rc

j

(−1)j E

{exp

[1k

(r − j) T β

]}. (5.16)

76


5.2.3 Entropies

Here, in this section shannon entropy is considered.

Theorem 5.2.2. Using Theorem 2 of Aljarrah et al.(2014), the Shannon entropy of T-Burr {Y}is given by

ηx = ηT + E (log fY (T )) + E(log Q′

Burr [FY (T )]). (5.17)

If Qx(u) is the quantile function of the Burr XII distribution.

QX(u) =[(1− u)−

1k − 1

] 1c. (5.18)

and

Q′X(u) =

1c k

(1− u)−1k−1

[(1− u)−

1k − 1

] 1c−1

(5.19)

Then we have the following shannon entropies for T -Burr{Lomax}, T -Burr{Log− logistic} and

T -Burr{Weibull} distributions are, respectively, given by

1. ηx = ηT + log(

1c k β2

)+ 1−3k

k E[log

(1 + T

β

)]+ (1− c) E (log X) ,

2. ηx = ηT +log(

β2

c k α2 β

)+2 (β−1)E (log T )+1−3k

k E[log

[1 +

(Tβ

)α]]+(1−c) E (log X) ,

3. ηx = ηT + log(

β2

c k

)2 (β − 1)E (log T ) + 1

kE(T β) + (1− c) E (log X) .

Proof:

If Y ∼ Lomax(1, β) having pdf and cdf, r(T ) = 1β

(1 + T

β

)−2and R(T ) = 1 −

(1 + T

β

)−1,

then

QY (r(T )) =

[(1 +

T

β

) 1k

− 1

] 1c

,

log Q′Y (Lomax) = log

(1

c k β

)+

(1k− 1

)E

[log

(1 +

T

β

)]+

(1− c

c

),

E

{log

[(1 +

T

β

) 1k

− 1

]}

log fY (T ) = log1β− 2 log

(1 +

T

β

).

77


Combining these results in Eq. (5.17), we have

ηx = ηT + log(

1c k β2

)+

1− 3k

kE

[log

(1 +

T

β

)]+ (1− c) E (log X) (5.20)

If Y ∼ log-logistic(α, β) having pdf and cdf, r(T ) = βα

(Tα

)β−1[1 +

(Tα

)β]−2

and R(T ) =

1−[1 +

(Tα

)β]−1

, then

QY [r(T )] =

(1 +

(T

α

)β) 1

k

− 1

1c

,

log Q′Y [r(T )] = log

(β

αβ c k

)+ (β − 1) (log T ) +

(1k− 1

){log

[1 +

(T

α

)]}

+(

1− c

c

)log

(1 +

(T

α

)β) 1

k

− 1

,

log fY (T ) = logβ

α+ (β − 1) log x− 2 log

[1 +

(T

α

)β]

.


ηx = ηT +log(

β2

c k α2 β

)+2 (β−1)E (log T )+

1− 3k

kE

[log

[1 +

(T

β

)α]]+(1−c) E (log X) .

(5.21)

If Y ∼ Weibull(1, β) having pdf and cdf, r(T ) = β e−T βand R(T ) = 1− e−T β

, then

QY [r(T )] =[e

xβ

k − 1] 1

c

,

log Q′Y [r(T )] = log

(β

c k

)+ (β − 1) log T +

T β

k+

(1− c

c

)log

(1 +

(T

α

)β) 1

k

− 1

,

+(

1− c

c

)log

[e

xβ

k − 1]

.

log fY (T ) = log β + (β − 1) log T − T β


ηx = ηT + log(

β2

c k

)2 (β − 1)E (log T ) +

1kE(T β) + (1− c) E (log X) . (5.22)

78


5.2.4 Mean Deviation

The mean deviations about mean and median for the T-Burr {Y} family are, respectively,

given by

δ1 = 2µF (µ)− 2 Ic (µ); δ2 = µ− 2 Ic (µ). (5.23)

FX(x) is given in Eq. (5.4), mean µ can be obtained from Eq. (5.13) for r = 1, their median

can be obtained from Remark 1(ii) by setting u = 0.5. The first incomplete moment Ic(s)

can be obtained as

Ic(s) =

s∫

0

xs fX(x)dx =

QY (FR(s))∫

0

QR(FY (w)) fT (w) dw. (5.24)

Using the result in Eq. (5.24) first incomplete moments for T-Burr{Lomax}, T-Burr{Log-

logistic} and T-Burr{Weibull} families of distributions can be obtained, respectively, as

Ic(s) =∞∑

j=0

1c

j

(−1)j

β[(1+sc)k−1]∫

0

(1 +

t

β

) 1k

(1−j)

fT (t) dt,

Ic(s) =∞∑

j=0

1c

j

(−1)j

α [(1+sc)k−1]1β∫

0

{1 +

(t

β

) 1β

} 1k

(1−j)

fT (t) dt,

Ic(s) =∞∑

j=0

1c

j

(−1)j

[k ln(1+sc)]1β∫

0

exp[T β(1− j)

k

]fT (t) dt.

5.3 Special Sub-Models

Some different distributions for T random variable are considered to generate special mod-

els. Three special models Gamma-Burr{log-logistic} , Dagum-Burr{Weibull} and Weibull-

Burr{Lomax} are considered. Some statistical properties of Weibull-Burr{Lomax} are

studied.

79


5.3.1 The Gamma-Burr{Log-logistic} (GaBLL) distribution.

If T follows the Gamma distribution having cdf F (t) = γ(a,t)Γ(a) , t > 0, where γ(a, t) =

t∫0

ta−1e−tdx is the lower gamma function, then cdf and pdf of GaBLL distribution are,


FX(x) = P(a, α [Burr(c,−k)]

1β

). (5.25)

where Burr(c,−k) = (1 + xc)k − 1. By setting α = 1 Eq. in (5.25), we have

FX(x) = P(a, [Burr(c,−k)]

1β

). (5.26)

and

fX(x) =burr(c, k)Γ(a)baβ

(Burr(c,−k))a−β

β exp[1b

((Burr(c,−k))

1β

)].

(a) (b)

0 1 2 3 4 5

0.0

0.1

0.2

0.3

0.4

x

pdf

c = 2 k = 0.5 a = 2 b = 2.2 β = 0.4c = 0.5 k = 0.3 a = 2 b = 0.2 β = 2c = 2.5 k = 0.5 a = 1.5 b = 0.8 β = 1.5c = 1.5 k = 0.5 a = 2 b = 0.5 β = 0.5

0 1 2 3 4 5

0.0

0.1

0.2

0.3

0.4

x

hrf

c = 2 k = 0.5 a = 2 b = 2.2 β = 0.4c = 0.5 k = 0.3 a = 2 b = 0.2 β = 2c = 1.5 k = 0.3 a = 1.5 b = 0.8 β = 1.2c = 1.5 k = 0.5 a = 2 b = 0.8 β = 0.6

Figure 5.1: Plots of (a) density and (b) hrf of GaBLL distribution

The density in Figure 5.1 (a) are the reversed J, symmetrical and left skewed and hrf in

Figure 5.1 (b) are decreasing, increasing and upsidedown bathtub.

5.3.2 The Dagum-Burr{Weibull} (DBW) distribution.

If T follows the Dagum distribution having cdf FT (t) = [1 + t−a]−b, t > 0, then the cdf

and pdf of DBW distribution are, respectively, given by

FX(x) =[1 + [k ln (1 + xc)]−

aβ

]−b(5.27)

80


By setting a = 1 in Eq. (5.27), we have

FX(x) =[1 + [k ln (1 + xc)]−

1β

]−b(5.28)

and

fX(x) =c k b xc−1

β (1 + xc)

[1 + [k ln (1 + xc)]−

1β

]−b−1

[k ln (1 + xc)]1β−1

(5.29)

(a) (b)

0 1 2 3 4

0.0

0.5

1.0

1.5

x

pdf

b = 5.7 β = 2 c = 2 k = 4b = 0.2 β = 2 c = 0.8 k = 3b = 2.5 β = 2 c = 3 k = 2.5b = 1.5 β = 0.6 c = 2 k = 1.1

0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

1.0

x

hrf

b = 5.7 β = 2 c = 2 k = 3b = 2 β = 2 c = 0.8 k = 3b = 5 β = 3 c = 2 k = 4b = 0.5 β = 3 c = 0.3 k = 8b = 5 β = 0.5 c = 3 k = 0.8

Figure 5.2: Plots of (a) density and (b) hrf of DBW distribution

The density in Figure 5.2 (a) are the reversed-J and left skewed and hrf in Figure 5.2(b)

are decreasing and upsidedown bathtub.

5.3.3 The Weibull-Burr{Lomax} (WBLx) distribution.

If T follows the Weibull distribution having cdf FT (t) = 1− e−a tb , then the cdf and pdf of

WBLx distribution are, respectively, given by

FX(x) = 1− exp[−a β

({(1 + xc)

kα − 1

})b]

(5.30)

By setting β = 1 and α = 1 in Eq. (5.30), we have

FX(x) = 1− exp[−a

({(1 + xc)k − 1

})b]

(5.31)

and

fX(x) = c k a b xc−1 (1 + xc)k−1{

(1 + xc)k − 1}b−1

× exp[−a

{(1 + xc)k − 1

}b]

(5.32)

81


(a) (b)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.0

0.2

0.4

0.6

0.8

1.0

x

pdf

a = 2 b = 0.7 c = 4 k = 0.2a = 2 b = 0.5 c = 0.5 k = 0.8a = 0.5 b = 2 c = 1.5 k = 0.7a = 0.1 b = 4 c = 2 k = 0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.5

1.0

1.5

2.0

x

hrf

a = 0.6 b = 1 c = 1 k = 1a = 1 b = 1.5 c = 0.9 k = 0.5a = 0.6 b = 1.5 c = 0.9 k = 1a = 1 b = 1.5 c = 0.8 k = 0.5a = 0.5 b = 1.5 c = 0.9 k = 1

Figure 5.3: Plots of (a) density and (b) hrf of WBLx distribution

The density in Figure 5.3 (a) are the reversed-J , left skewed, right skewed and symmetrical

and hrf in Figure 5.3 (b) are increasing, decreasing, upsidedown bathtub and constant. qf

for WBLx can be obtained form the Remark 1(ii)

QX(u) =

(1 +

[−1

aln(1− u)

] 1b

) 1k

− 1

1c

.

Mode of WBLx can be obtained form the Eq. (5.12)

d

dxfT (x) =

c− 1x

+ (k − 1)cxc−1

1 + xc+ (b− 1) c k

xc−1 (1 + xc)k−1

{(1 + xc)k − 1}

−a b c k({

(1 + xc)k − 1})b−1

(1 + xc)k−1xc−1.

rth Moment of WBLx can be obtained form the Eq. (5.14)

E(Xr) =∞∑

j=0

∞∑

i=0

(−1)j

rc

j

r−jk

i

[γ

(1 +

i

b, a

)+ Γ

(1 +

r−jk − i

b, a

)], (5.33)

where γ (a, x) =x∫0

ta−1e−t dt and Γ (a, x) =∞∫x

ta−1e−t dt are the lower and upper incom-

plete gamma functions.

From Eq.(5.20) the Shannon entropy of the WBLx is given by

ηX = ηT − log(c k) +(

1 + k

k

)E (log(1 + T )) + (1− c)E (log X) ,

82


where ηT = log(a b) +(1 + 1

b

)ξ − a and ξ is the Euler gamma constant and

E (log(1 + T )) =b− 1

blog a. exp(−a)− EI(−a) +

∞∑

n=1

(−1)n+1anb

nΓ

(−n

b+ 1, a

)

+∞∑

n=1

(−1)n

nanb

γ(n

b+ 1, a

).

[Aljarrah et al.(2015)]

EI(x) =x∫

−∞t−1etdt is the exponential integral (abramowitz and Steyum 1972). and E (log X) =

limx→0

ddxE(xr), where E(Xr) is given in (5.33).

If x1, x2, ..., xn be a random sample from WBLx distribution, then the log-likelihood func-

tion for the vector of parameters Θ = (a, b, c, k)T is

l(Θ) = n log(a b c k) + (c− 1)n∑

i=1

log xi + (k − 1)n∑

i=1

log(1 + xci )

+ (b− 1)n∑

i=1

log{

(1 + xci )

k − 1}− a

n∑

i=1

{(1 + xc

i )k − 1

}b


Ua =n

a−

n∑

i=1

{(1 + xc

i )k − 1

}b,

Ub =n

b+

n∑

i=1

log{

(1 + xci )

k − 1}− a

n∑

i=1

{(1 + xc

i )k − 1

}blog

{(1 + xc

i )k − 1

},

Uc =n

c+

n∑

i=1

log xi + (k − 1)n∑

i=1

[xc

i log xi

1 + xci

]+ (b− 1)

n∑

i=1

[k (1 + xc

i )k−1 xc

i log xi

(1 + xci )k − 1

]

− a b

n∑

i=1

{(1 + xc

i )k − 1

}b−1k (1 + xc

i )k−1 xc

i log xi,

Uk =n

k+

n∑

i=1

log(1 + xci ) + (b− 1)

n∑

i=1

[(1 + xc

i )k log(1 + xc

i )(1 + xc

i )k − 1

]

− a bn∑

i=1

{(1 + xc

i )k − 1

}b−1(1 + xc

i )k log(1 + xc

i ).

Setting Ub, Ua, Uk and Uc equal to zero and solving these equations simultaneously yields


83


5.4 Simulation and application

Here, in this section simulation and application are given for WBLx distribution.

5.4.1 Simulation

We study the performance of the MLE of WBLx distribution by using different sizes (n=100,200,

500), 1000 samples are simulated for the true parameters values I: c= 2 k= 0.5 a= 1 b= 1 and

II : c= 3 k= 1.5 a= 1.5 b= 0.5 in order to obtain average estimates (AEs), bias and mean

square errors (MSEs) of the parameters, they are listed in Table 5.2. The small values of the

biases and MSEs, and MSE decreases as the sample size increases The results indicate that

the maximum likelihood method performs quite well for estimating the model parameters

of the proposed distribution.

Table 5.2: Estimated AEs, biases and MSEs of the MLEs of parameters of WBLx distribution

based on 1000 simulations for n=100, 200 and 500.

I II

n parameters A.E Bias MSE A.E Bias MSE

100 c 2.752 0.752 4.622 4.571 1.571 11.839

k 0.554 0.054 0.059 1.844 0.344 0.955

a 1.385 0.385 1.710 1.663 0.163 1.432

b 1.074 0.074 0.439 0.557 0.057 0.202

200 c 2.298 0.298 1.185 4.021 1.021 6.407

k 0.538 0.038 0.033 1.618 0.118 0.311

a 1.380 0.380 1.588 1.503 0.043 0.356

b 1.041 0.041 0.244 0.546 0.046 0.122

500 c 2.046 0.246 1.128 3.680 0.680 4.757

k 0.501 0.001 0.017 1.610 0.110 0.146

a 1.020 0.300 0.895 1.418 0.003 0.246

b 1.038 0.038 0.153 0.527 0.037 0.118

5.4.2 Application

Here, applications on two data sets complete (uncensored) data set and for censored data

set are given, to show the performance of the WBLx distribution.

84


5.4.3 Complete data set 6: Diameter-Thickness

The WBLx distribution is used for real data sets. We fit the WBLx, Kumaraswamy Burr(Kw-

Bu), Beta Burr(B-Bu), Beta exponential(B-Exp), Burr and Weibull to a data set. The data set

of 50 observations, hole diameter and sheet thickness are 9 mm and 2 mm respectively.

Hole diameter readings are taken on jobs with respect to one hole, selected and fixed as

per a predetermined orientation. The data set is given by Ratan (2011).

The summary statistics from the first data set are: x = 0.152, s = 0.0061, γ1 = 0.0061 and

γ2 = 2.301226, where γ1 and γ2 are the sample skewness and kurtosis respectively.

Table 5.3: MLEs and their standard errors (in parentheses) for Data set 6

Distribution a b c k α β

WBLx 0.565 0.807 1.663 19.342 - -

(0.82) (0.41) (1.11) (22.99)

Kw-Bu 0.227 11.522 8.340 - 39.720

(0.028) (3.658) (0.007) - (0.999) -

B-Bu 27.607 9.738 5.070 - 0.029 -

(87.432) (1.951) (10.925) - (0.032)

B-Exp 2.667 18.006 - - - 0.9321

0.5042 99.87 - - - 4.96

Burr - - 2.043 37.66 -

- - (0.231) - (14.540) -

Weibull 34.45 2.002 - - - -

(13.755) (0.235) - - - -

Table 5.4 shows that Weibull-Burr{Lomax} (W-B{Lx}) among the Beta Burr(B-Bu), Ku-

maraswamy Burr (Kw-Bu) , Beta Exponential(B-Exp), Burr and Weibul distributions gives

better fit. The estimated pdfs and cdfs of the W-B{Lx} model and other models are dis-

played in Figure 5.4.

85


Table 5.4: The Value, W*, A*, KS, P-Value values for data Set 6

Dist −` W* A* KS P − V alue

WBLx 59.62026 0.1103664 0.6764127 0.1269 0.3969

Kw-Bu 57.88482 0.1976216 1.119699 0.1597 0.1558

B-Bu 54.90359 0.3194159 1.75434 0.2073 0.02716

B-Exp 54.62055 0.3224291 1.777851 0.2098 0.02455

Burr 57.10991 0.2166066 1.227761 0.1689 0.1153

Weibull 57.30266 0.212311 1.203196 0.1691 0.1144

5.4.4 Censored data set 7: Remission-Times

Here, application on W-B{Lx} model on censored data set is given. The W-B{Lx} is com-

pared with Kw-Bu and B-Bu distributions. The data below are remission times, in weeks,

for a group of 30 patients with leukemia who received similar treatment, quoted in Jerlald

F(2003).

Consider a data set D = (x, r), where x = (x1, x2, ..., xn)T are the observed failure times

and ri = (r1, r2, ..., rn)T are the censored failure times. The ri is equal to 1 if a fail-

ure is observed and 0 otherwise. Suppose that the data are independently and identi-

cally distributed and come from a distribution with pdf given by equation (5.32). Let

Θ = (c, k, a, b)T denote the vector of parameters. The likelihood of Θ can be written as

l(D; Θ) =n∏

i=1

[f(xi; Θ)]ri [1− F (xi; Θ)]1−ri (5.34)

The log likelihood for W-B{Lx} is

l = log K +n∑

i=1

ri

[log(c k a b) + (c− 1) log xi + (k − 1) log(1 + xc

i )

+ (b− 1) log{

(1 + xc)k − 1}− a

{(1 + xc)k − 1

}b]

+n∑

i=1

(1− ri)

×[−a

{(1 + xc)k − 1

}b]

(5.35)

The log likelihood function can be maximized numerically to obtained the MLEs. There

are various routines available for numerical maximization of l. We use the routine optim in

86


the R software. It is observed that AIC and BIC statistics WBLx are minimum as compare

Table 5.5: MLEs and their standard errors for Data set 7

Model Parameters MLE Standard error Log-Likelihood AIC BIC

WBLx c 1.2902 0.7573 -108.2892 224.5785 230.1832

k 0.0675 0.0600

a 9.2729 19.2363

b 1.9982 0.4593

Kw-Bu c 1.6530 0.3119 -111.7468 231.4935 237.0983

k 15.7654 14.8965

a 12.3872 9.3006

b 0.0051 0.0023

B-Bu c 0.2236 0.2691 -108.3125 224.6249 230.2297

k 2.9653 7.0795

a 0.6207 0.2738

b 26.4381 30.1542

to Kw-Bu and B-Bu distribution.

(a) (b)

x

Den

sity

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

01

23

45

W−Bu{Lx}B−BuKw−Bu

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

0.0

0.2

0.4

0.6

0.8

1.0

x

cdf

W−Bu{Lx}B−BuKw−Bu

Figure 5.4: Estimated (a) pdfs and (b) cdfs for data set 6.

87


0 20 40 60 80

0.0

0.2

0.4

0.6

0.8

1.0

Empirical and theoretical CDFs

x

CD

F

W−Bu{Lx}Kw−BuB−Bu

Figure 5.5: Plots of estimated cdf for censored data set 7.

5.5 Conclusions and Results

In this chapter, T-Burr{Y} class of distributions and three new distributions Gamma-Burr{Log-

logistic}, Dagum-Burr{Weibull} and Weibull-Burr{Lomax} are introduced. The explicit

expressions for their qf, mode, rth moment and mean deviations and Shannon entropy are

studied. A special model is discussed in detail. Application is carried out on proposed

family through three special sub models on the real life data sets on censored and com-

plete samples to check the usefulness of the family. We conclude that our proposed family

provide better results than other competing models.

88

Chapter 6

Kumaraswamy Odd Burr XII Family

of distributions

6.1 Introduction

The cdf and pdf of Kumaraswamy G family are, respectively, given by

F (x) =

G(x)∫

0

a b xa−1 (1− xa)b−1 dx

= 1− (1−Ga(x))b (6.1)

and

f(x) = a b g(x) Ga−1(x) (1−Ga(x))b−1 (6.2)

Using the odd Burr G family (Alizadeh et al.,2016), the cdf and pdf are, respectively,

given by

G(x) = 1−{

1 +(

R(x)1−R(x)

)c}−k

(6.3)

and

g(x) = c k g(x)Rc−1(x)

(1−R(x))c+1

{1 +

(R(x)

1−R(x)

)c}−k−1

(6.4)

89


Let Bc,k(x) = G(x) and bc,k(x) = g(x) for convenience. Using Eqs. (6.3) and (6.4) in Eqs.

(6.1) and (6.2), we have the cdf and pdf of Kumaraswamy odd Burr G (KOBG) famly are,


F (x) = a b

Bc,k(x)∫

0

ta−1 (1− ta)b−1 dt

or

F (x) = 1− (1− {Bc,k(x)}a)b (6.5)

and

f(x) = a b bc,k(x) {Bc,k(x)}a−1 (1− {Bc,k(x)}a)b−1 . (6.6)

The qf Q(u) can be determined by inverting Eq. (6.5), we have

QX(u) = R−1

[(1− z)−

1k − 1

] 1c

1 +[(1− z)−

1k − 1

] 1c

, (6.7)

where z =[1− (1− u)

1b

] 1a and U ∼ Unifrom(0, 1).

The hrf of Eq. (6.6), is given by

h(x) =a b bc,k(x) {Bc,k(x)}a−1

1− {Bc,k(x)}a

6.2 Infinite mixture representation

Here, infinite mixture representation of the KOBG distribution are given.

Theorem 6.2.1. If X ∼ KOBG(a, b, c, k), then we have the following approximations.

A: If a, b > 0 and c, k > 0 are the real non-integer values, then we have following infinite mixture

representation of cdf of KOBG distribution is

F (x) =∞∑

q=0

wq Hq(x), (6.8)

90


where Hq(x) = Rq(x) represents the exp-R distribution with power parameter q. The coefficients

are defined as

wq =∞∑

j,i=0

∞∑

m,n=0

∞∑

s=0

b

j

aj

i

kj + m− 1

m

cm + n− 1

n

× cm + n

s

s

q

(−1)s+q+m+i+j (6.9)

B: Infinite mixture representation of pdf of KOBG distribution is

f(x) =∞∑

q=0

wq hq+1(x), (6.10)

where wq are defined in Eq. (6.9)

Proof:

If b > 0 is a real number, then we have following series expansions

(1− z)−b =∞∑

j=0

b + j − 1

j

zj (6.11)

(1− z)b =∞∑

j=0

b

j

(−1)j zj (6.12)

Using Eqs.(6.11) and (6.12) in the Eq. (6.5) we have

F (x) =∞∑

q=0

wq Hq(x),

where wq is given in (6.9). Hq(x) is the exp-G distribution of the base line densities with q

as power parameter. Eq. (6.10) can easily obtained by simple derivative of Eq. (6.8)

6.3 General properties

Here, the rth moment, mth Incomplete moment, moment generating function and mean

deviations of the KOBG family of distribution are studied.

91


6.3.1 Moments

The rth moment of the KOBG family of distributions can be obtained by using the follow-

ing expression

E(Xr) =∞∑

q=0

wq

∞∫

0

xr hq+1(x)dx, (6.13)

where wq are defined in Eq. (6.9).

The mth incomplete moment of the KOBG family of distributions can be obtained by using

the following expression

µm(x) =∞∑

q=0

wq T ′m(x), (6.14)

where T ′m(x) =x∫0

xr hq+1(x)dx.

The moment generating function of the KOBG family of distributions is obtained as

MX(t) =∞∑

q=0

wq Mq+1(t), (6.15)

where Mq+1(t) =∞∫0

et x hq+1(x)dx.

The mean deviations of the KOBG family of distributions about the mean and median,

respectively, can be obtained as

Dµ = 2µF (µ)− 2µ1(µ) (6.16)

DM = µ− 2µ1(M) (6.17)

where µ = E(X), can be obtained from Eq. (6.13), M = Median(X) can be obtained from

Eq. (6.7), F (µ) can be calculated easily from Eq. (6.5) and µ1(.) can be obtained from Eq.

(6.14) by setting m = 1.

6.4 Entropies

Here, we will consider only two entropies, renyi and shannon.

92


Theorem 6.4.1. If x ∼ KOBG(a, b, c, k), then we have the following approximations.

A: If δ > 0 and a, b, c, k > 0 be the real non-integer values, then we have the following expression

for Rayni entropy

IR =1

1− δlog

∞∑

l,m=0

Vl,m(δ)

∞∫

0

r(x; δ) R(x;m + c(l + δ)− δ) dx

where r(x; δ) = rδ(x) and R(x; m + c(l + 1)− 1) = Rm+c(l+1)−1(x) represents the pdf and cdf of

the baseline distribution with power parameter δ and m + c(l + 1)− 1. The above integral depends

only on the baseline distribution. The coefficients are defined as

Vl,m(δ) =∞∑

j,i=0

δ(b− 1)

j

a(j + δ)− δ

i

k(i + δ) + δ + l − 1

l

c(l + δ) + δ + m− 1

m

(−1)i+j

B: If g(x) be the density of the KOBG family of distributions, then we have shannon entropy of

KOBG family as

ηx = M− (c + 1)E {log [1−R(x)]} − (c− 1)E [log R(x)]

−(k + 1)∞∑

j=1,i=0

ai,j(c) E(Rc j+i(x)

)− (a− 1)∞∑

i,l=0

bi,l(c, k) E(Rc i+l(x)

)

−(b− 1)∞∑

l,m=0

al,m(a, c, k) E(Rc l+m(x)

)− E (log r(x))

where r(x), R(x) represents the pdf and cdf of the base line densities. The above expectations only

depends on the baseline densities. The coefficients are defined as

ai,j(c) =(−1)j+1

j

c j + i− 1

i

(6.18)

bi,l(c, k) =∞∑

j=1

(−1)i+1

j

k j + i− 1

i

c i + l − 1

l

(6.19)

al,m(a, c, k) =∞∑

j=1,i=0

(−1)i+l+1

j

a j

i

k i + l − 1

l

c l + m− 1

m

(6.20)

93


Proof A:

The renyi entropy of KOBG family of distribution is

IR =1

1− δlog

∞∫

0

[a b bc,k(x) {Bc,k(x)}a−1 (1− {Bc,k(x)}a)b−1

]δdx (6.21)

Using Eqs. (6.11) and (6.12), we have

f δ(x) = (a b c k)δ∞∑

i,j

δ(b− 1)

j

a(j + δ)− δ

i

∞∑

l,m

k(j + δ) + δ + l − 1

l

× c(l + δ) + δ + m− 1

m

(−1)i+j g(x; δ) G(x; m + c(l + δ)− δ)

Proof B:

Using following series expansions

log(1− x) = −∞∑

j=1

xj

j(6.22)

and

log(1 + x) =∞∑

j=1

(−1)j+1

jxj . (6.23)

The shannon entropy of KOBG family of distribution is

ηx = M−E (log r(x))− (c− 1) E (log R(x)) + (c + 1)E(log R(x)

)

+(k + 1)E(

log 1 +[

R(x)1−R(x)

]c)− (a− 1)E

(log 1−

{1 +

[R(x)

1−R(x)

]c}−k)

−(b− 1) log

{1−

[1−

{1 +

[R(x)

1−R(x)

]c}−k]a}

. (6.24)

Using Eqs.(6.23) and (6.12), we have

log{

1 +[

R(x)1−R(x)

]c}=

∞∑

j=1,i=0

(−1)j+1

j

c j + i− 1

i

Rc j+i(x).

Using Eqs.(6.22), (6.11) and (6.12), we have

log

[1−

{1 +

[R(x)

1−R(x)

]c}−k]

=∞∑

j=1

∞∑

i,l=0

(−1)i+1

j

k j + i− 1

i

c i + l − 1

l

Rc i+l(x).

94


Using Eqs. (6.22), (6.11) and (6.12), we have

log

{1−

[1−

{1 +

[R(x)

1−R(x)

]c}−k]a}

=∞∑

j=1

∞∑

i,l,m=0

(−1)i+l+1

j

a j

i

k i + l − 1

l

× c l + m− 1

m

Rc l+m(x).

Combining all these results in Eq. (6.24), we have

ηx = M− (c + 1)E {log [1−R(x)]} − (c− 1)E [log R(x)]

−(k + 1)∞∑

j=1,i=0

ai,j(c) E(Rc j+i(x)

)− (a− 1)∞∑

i,l=0

bi,l(c, k) E(Rc i+l(x)

)

−(b− 1)∞∑

l,m=0

al,m(a, c, k) E(Rc l+m(x)

)− E (log r(x)) ,

where r(x), R(x) are the of pdf and cdf baseline distribution. The coefficients are defined

in Eqs.(6.19), (6.20) and (6.20).

6.5 stochastic ordering, moments ofresidual and reversed resid-

ual life

Here, the stochastic ordering, residual and reversed residual life.


Let X1 ∼ KOBG(a, b1, c, k) and X2 ∼ KOBG(a, b2, c, k) with density functions

f(x) = a b1 bc,k(x) Ba−1c,k (x)

[1−Ba

c,k(x)]b1−1

g(x) = a b2 bc,k(x) Ba−1c,k (x)

[1−Ba

c,k(x)]b2−1

Now we consider the ratio

f(x)g(x)

=b1

b2

[1−Ba

c,k(x)]b1−b2

Taking derivative with respect to x, we have

d

dx

f(x)g(x)

= ab1

b2(b1 − b2) bc,k(x) Ba−1

c,k (x)[1−Ba

c,k(x)]b1−b2−1

95


From the above expression, we observe that if b1 < b2 ⇒ ddx

f(x)g(x) < 0, then this implies that


6.5.2 Moments of Residual and Reversed residual life

Theorem 6.5.1. If n is an integer value n > 1 and x > t, and X ∼ KOBG family of distribu-

tions, then we have the following approximations.

Moments of Residual Life

mn(t) =1

R(t)

∞∑

q=0

wq

∞∑

m=0

n

m

(−t)m

∞∫

t

xn−m hq+1(x) dx (6.25)

Moments of Reversed Residual Life

Tn(t) =1

F (t)

∞∑

q=0

wq

∞∑

m=0

n

m

(t)n−m (−1)m,

t∫

0

xm hq+1(x) dx. (6.26)

where hq+1(x) is the exp-R distribution of the base line densities, with q + 1 the power parameter.

Proof:

The nth moment of the residual life of KOBG family, is given by

E [(x− t)n|x > t] = mn(t) =1

R(t)

∞∫

t

(x− t)n f(x) dx.

If n is an integer value, we have following series expansion

(a− b)n =∞∑

j−0

n

j

(−1)j bj (−1)n−j an−j , where |a| < b. (6.27)

Using the series expansion in Eq. (6.27), and infinite mixture representation in Eq.(6.8). By

changing the order of integration and summation, we have

mn(t) =1

R(t)

∞∑

q=0

wq

∞∑

m=0

n

m

(−t)m

∞∫

t

xn−m hq+1(x) dx.

where hq−1(x) is the exp-R distribution of the base line densities, with q − 1 as the power

parameter. The coefficients wq are defined in Eq. (6.9). The result in Eq. (6.26) can be

obtained easily by following the steps used to obtain the result in Eq. (6.25).

96



Here, the expression of the ith order statistics as the infinite mixture representation of

baseline pdf and cdf.

Theorem 6.6.1. A. If n is an integer value and for i = 1, 2, ..., n and X1, X2, ..., Xn be identi-

cally independently distributed random variables, then the density of ith order statistics is

fi:n(x) =n−i∑

j=0

∞∑

p,q=0

mj(p, q)hp+q(x), (6.28)

where

mj(p, q) =

n− i

j

(−1)j wp ej+i−1:q(p + 1)

β(i, n− i + 1) (p + q + 1)(6.29)

hp+q(x) = (p + q + 1) g(x) Gp+q(x) are the exp-G densities with power parameter ”p+q”.

B. If j ≥ 1 is an integer value, then we have the following probability weighted moments of the

KOBG family of distributions.

E(xsi:n) = s

n∑

j=n−i+1

(−1)j−n+i−1

j − 1

n− i

n

j

Ij(s),

where Ij(s) =j∑

m=0

j

m

(−1)m

∞∑

q=0

em:q

∞∫

−∞xs−1

i Gq(xi)dx. (6.30)

and Gq(xi) is the exp-G distribution of the base line densities, with ”q” as power parameter.

Proof of A: If n ≥ 1 is an integer value then, we have following power series expansion

(Gradshtegn and Ryzhik, 2000).[ ∞∑

k=0

ak xk

]n

=∞∑

k=0

ak:n xk, (6.31)


m a0

m∑k=1


The expression for ith order statistics is

fi:n(x) =1

β(i, n− i + 1)g(x) Gi−1(x) [1−G(x)]n−i

97


Using Eq. (6.11) we obtain

fi:n(x) =1

β(i, n− i + 1)

n−i∑

j=0

n− i

j

(−1)if(x)[F (x)]i+j−1

Using the infinite mixture representation in Eqs. (6.8), (6.10) and series in Eq. (6.31), we

have

fi:n(x) =n−i∑

j=0

∞∑

p,q=0

mj(p, q) hp+q(x)

where the coefficients are defined in Eq. (6.29).

Proof of B:

Science Ij(s) =∞∫−∞

xs−1i {1− F (xi)}j and if j ≥ 1 then using Eq. (6.11), then we have

{1− F (xi)}j =j∑

m=0

j

m

(−1)m Fm(xi). (6.32)

Using Eq. (6.31), we obtain

{1− F (xi)}j =j∑

m=0

j

m

(−1)m

∞∑

q=0

eq:m Gq(xi). (6.33)

Substituting Eqs. (6.32) and (6.33) in Ij(s), we have

Ij(s) =j∑

m=0

j

m

(−1)m

∞∑

q=0

em:q

∞∫

−∞xs−1

i Gq(xi) dx.

6.7 Estimation

Here, the maximum likelihood estimates (MLEs) of the model parameters of the KOBG

family for complete and censored samples are given. Let x1, x2, ..., xn be a random sample

of size n from the KOBG family of distributions.

98



The log-likelihood function for the vector of parameters Θ = (a, b, c, k)T is

`(Θ) = n log(a b c k) +n∑

i=1

log [r(xi)] + (c− 1)n∑

i=1

log [R(xi)]− (c + 1)n∑

i=1

log[R(xi)

]

−(k + 1)n∑

i=1

log[1 +

(R(xi)R(xi)

)c]+ (a− 1)

n∑

i=1

log [Bc,k(xi)]

+(b− 1)n∑

i=1

log[1−Ba

c,k(xi)],

where Bc,k(xi) = 1−{

1 +(

R(x,ξ)1−R(x,ξ)

)c}−k.

The components of the score vector are given by

Ua =n

a+

n∑

i=1

log [Bc,k(xi)]− (b− 1)n∑

i=1

[Ba

c,k(xi) log Bc,k(xi)1−Ba

c,k(xi)

],

Ub =n

b+

n∑

i=1

log[1−Ba

c,k(xi)],

Uc =n

c+

n∑

i=1

log [R(xi)]−n∑

i=1

log[R(xi)

]− (k + 1)n∑

i=1

(R(xi)R(xi)

)clog

(R(xi)R(xi)

)

1 +(

R(xi)R(xi)

)c

+(a− 1)n∑

i=1

[∂∂cBc,k(xi)Bc,k(xi)

]− (b− 1)

n∑

i=1

[aBa−1

c,k (xi) ∂∂cBc,k(xi)

1−Bac,k(xi)

],

Uk =n

k−

n∑

i=1

log[1 +

(R(xi)R(xi)

)c]− (a− 1)

n∑

i=1

[∂∂kBc,k(xi)Bc,k(xi)

]

+(b− 1)n∑

i=1

[aBa−1

c,k (xi) ∂∂kBc,k(xi)

1−Bac,k(xi)

].

Setting Ua, Ub, Uc and Uk equal to zero and solving these equations simultaneously yields


6.7.2 Estimation of parameters in case of censored complete samples


the likelihood function for type II censoring is

l(xi; Θ) = A

[r∏

i=1

f(xi; Θ)

]×

(1− F (xi; Θ)

)n−r, (6.34)

99


where f(.) and 1 − F (.) are the pdf and sf of KOBG family; X = (x1, x2, ..., xr) , Θ =

(θ1, θ2, ..., θn) and A=Constant. Inserting Eqs. (6.6) and (6.5) in Eq. (6.34), we have

l(xi; Θ) = A

[r∏

i=1

a b bc,k(x) {Bc,k(x)}a−1 (1− {Bc,k(x)}a)b−1

]×

((1− {Bc,k(x)}a)b

)n−r.

(6.35)

The log-likelihood function will be

`(xi; Θ) = log A + n log(a.b) +r∑

i=1

log bc,k(x) + (a− 1)r∑

i=1

log Bc,k(x)

+(b− 1)r∑

i=1

log (1− {Bc,k(x)}a) + b(n− r) log (1− {Bc,k(x)}a) .

The components of score vector are given by

Ua =n

a+

r∑

i=1

log Bc,k(xi)− (b− 1)r∑

i=1

[{Bc,k(xi)}a log {Bc,k(xi)}1− {Bc,k(xi)}a

]

+b(n− r)[{Bc,k(xr)}a log {Bc,k(xr)}

1− {Bc,k(xr)}a

],

Ub =n

b+

r∑

i=1

log (1− {Bc,k(xi)}a) + (n− r) log (1− {Bc,k(xr)}a) ,

Uc =n

c+

r∑

i=1

[dd cbc,k(xi)bc,k(xi)

]− a(b− 1)

r∑

i=1

[{Bc,k(xi)}a−1 d

d cBc,k(xi)1−Ba

c,k(xi)

]

+(a− 1)r∑

i=1

[dd cBc,k(xi)Bc,k(xi)

]− a b (n− r)

r∑

i=1

[{Bc,k(xr)}a−1 d

d cBc,k(xr)

1− dd cB

ac,k(xr)

],

Uk =r∑

i=1

[d

d kbc,k(xi)bc,k(xi)

]+ (a− 1)

r∑

i=1

[d

d kBc,k(xi)Bc,k(xi)

]

−a(b− 1)r∑

i=1

[Ba−1nc,k(xi) d

d kBc,k(xi)1−Ba

c,k(xi)

]− a b (n− r)

[Ba−1nc,k(xr) d

d kBc,k(xr)1−Ba

c,k(xr)

].

Setting Ua, Ub, Uc and Uk equal to zero and solving these equations simultaneously yields


6.8 Special Sub Models

Here, some special models of KOBG family are considered with their plots of density and

haard rate function.

100


6.8.1 The Kumaraswamy odd Burr-Frechet (KOBFr) distribution.

If Frechet distribution is baseline distribution having pdf and cdf, r(x) = αβx−β−1e−αx−β

and cdf R(x) = e−αx−β, then the cdf and pdf of KOBFr distribution are, respectively, given

by

F (x) = 1−{

1−(

1−{

1 +(eα x−β − 1

)c}−k)a}b

. (6.36)

and

f(x) = a b c k α β x−β−1

(e−α x−β

)c

(1− e−α x−β

)c+1

{1 +

(eα x−β − 1

)c}−k−1

×(

1−{

1 +(eα x−β − 1

)c}−k)a−1 {

1−(

1−{

1 +(eα x−β − 1

)c}−k)a}b−1

.

(i) If a = b = 1 in Eq. (6.36), then KOBG distribution becomes OBFr distribution, (ii) if

a = b = c = 1 in Eq. (6.36), then KOBG distribution becomes OLxFr distribution, (iii) if

a = b = k = 1 in Eq. (6.36), then KBG distribution becomes OLLFr distribution, (iv) If

a = b = c = k = 1 in Eq. (6.36), then KOBG distribution becomes Frechet distribution.

Figure 6.1 gives the plots of density and hrf of KOBG distribution. In Figure 6.1 (a) pdf are

right-skewered, left-skewered, symmetrical and reversed-J. The hrf in Figure 6.1 (b) are

increasing, decreasing, bathtub and upside-down bathtub.

6.8.2 The Kumaraswamy odd Burr-Lomax (KOBLx) distribution.

If Lomax distribution is base line distribution having pdf and cdf, r(x) = αβ

(1 + x

β

)−α−1

and R(x) = 1−(1 + x

β

)−α, then cdf and pdf of KOBLx distribution are, respectively, given

by

F (x) = 1−{

1−(

1−{

1 +[(

1 +x

β

)α

− 1]c}−k

)a}b

. (6.37)

101


(a) (b)

0 1 2 3 4

0.0

0.5

1.0

1.5

x

pdf

a = 1 b = 0.5 c = 2 k = 5 α = 2 β = 2a = 0.8 b = 2.5 c = 2 k = 2 α = 2 β = 0.1a = 0.8 b = 0.5 c = 3 k = 5 α = 3 β = 2a = 0.6 b = 0.5 c = 1 k = 1.5 α = 1 β = 2

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.5

1.0

1.5

2.0

x

hrf

a = 0.2 b = 0.8 c = 0.2 k = 0.4 α = 3 β = 2a = 0.8 b = 2 c = 2 k = 2 α = 2 β = 0.1a = 3 b = 4 c = 3 k = 3 α = 3 β = 1.3a = 0.8 b = 0.7 c = 3 k = 5 α = 2 β = 0.8a = 0.05 b = 0.7 c = 3 k = 5 α = 2 β = 0.8

Figure 6.1: Plots of (a) density and (b) hrf for KwOBuFr distribution for different parameter

values.

and

f(x) = a b c kα

β

(1 +

x

β

)−α−1

(1−

(1 + x

β

)−α)c−1

((1 + x

β

)−α)c+1

{1 +

[(1 +

x

β

)α

− 1]c}−k−1

×(

1−{

1 +[(

1 +x

β

)α

− 1]c}−k

)a−1 {1−

(1−

{1 +

[(1 +

x

β

)α

− 1]c

}−k)a}b−1

.

(i) If a = b = 1 in Eq. (6.37), then KOBLx distribution becomes OBLx distribution, (ii) if

a = b = c = 1 in Eq. (6.37), then KOBLx distribution becomes OLxLx distribution, (iii) if

a = b = k = 1 in Eq. (6.37), then KOBLx distribution becomes OLLLx distribution, (iv) If

a = b = c = k = 1 in Eq. (6.37), then KOBLx distribution becomes Lomax distribution.

Figure 6.2 shows the plots of density and hrf of KOBLx distribution. The pdf in Figure 6.2

(a) are right-skewered, left-skewered, symmetrical and reversed-J. The hrf in Figure 6.2 (b)

are increasing, decreasing and upside-down bathtub.

102


(a) (b)

0.0 0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

2.0

x

pdf

a = 0.1 b = 0.1 c = 1 k = 2 α = 0.2 β = 2a = 3 b = 2 c = 1 k = 3 α = 0.2 β = 0.3a = 6 b = 3 c = 1.5 k = 2 α = 8 β = 4.5a = 2.5 b = 2 c = 2.5 k = 2.5 α = 8 β = 5

0.0 0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

2.0

x

hrf

a = 0.5 b = 0.5 c = 1 k = 2 α = 0.5 β = 2a = 3 b = 2 c = 1 k = 3 α = 0.2 β = 0.3a = 2 b = 0.5 c = 1.5 k = 3 α = 0.2 β = 0.5a = 2 b = 2 c = 2 k = 2 α = 2 β = 0.6a = 5 b = 5 c = 1 k = 2 α = 5 β = 2

Figure 6.2: Plots of (a) density and (b) hrf for KOBLx distribution with different parametric

values.

6.8.3 The Kumaraswamy odd Burr-Dagum distribution.

If Dagum distribution is baseline distribution having pdf and cdf,

r(x) = α pβ

(xβ

)−α−1[1 +

(xβ

)−α]−p−1

and R(x) =[1 +

(xβ

)−α]−p

, then the cdf and pdf

of KOBD distribution are, respectively, given by

F (x) = 1−

1−

1−

1 +

[{1 +

(x

β

)−α}p

− 1

]−c

−k

a

b

. (6.38)

103


and

f(x) = a b c kα p

β

(x

β

)−α−1[1 +

(x

β

)−α]−p−1

([1 +

(xβ

)−α]−p

)c−1

(1−

[1 +

(xβ

)−α]−p

)c+1

×1 +

[{1 +

(x

β

)−α}p

− 1

]−c

−k−1

×

1−

1 +

[{1 +

(x

β

)−α}p

− 1

]−c

−k

a−1

×

1−

1−

1 +

[{1 +

(x

β

)−α}p

− 1

]−c

−k

a

b−1

.

(i) If a = b = 1 in Eq. (6.38), then KOBD distribution becomes OBD distribution, (ii) if

a = b = c = 1 in Eq. (6.38), then KOBD distribution becomes OLxD distribution, (iii) if

a = b = k = 1 in Eq. (6.38), then KOBD distribution becomes OLLD distribution, (iv) If

a = b = c = k = 1 in Eq. (6.38), then KOBD distribution becomes Dagum distribution.

Figure 6.3 shows the plots of density and hrf of KOBD distribution. The pdf in Figure 6.3

(a) are right-skewered, symmetrical and reversed-J. The hrf in Figure 6.3 (b) are increasing,

decreasing and upside-down bathtub.

6.8.4 The Kumaraswamy odd Burr-Gompertz (KOBGo) distribution.

If Gompertz distribution is baseline distribution having pdf and cdf

r(x) = α eβ x exp{−α

β

[eβ x − 1

]}and cdf R(x) = 1 − exp

{−α

β

[eβ x − 1

]}. where β >

0 , α > 0, then the cdf and pdf of KOBGo distribution are, respectively, given by

F (x) = 1−{

1−(

1−{

1 +[exp

{α

β

[eβ x − 1

]}− 1

]c}−k)a}b

. (6.39)

104


(a) (b)

0.0 0.5 1.0 1.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

x

pdf

a = 1 b = 2 c = 2 k = 2 α = 2 β = 0.5 p = 3a = 3 b = 2 c = 1 k = 3 α = 0.2 β = 0.3 p = 1a = 6 b = 3 c = 0.5 k = 2 α = 2 β = 0.5 p = 1a = 2 b = 2 c = 1.5 k = 2 α = 3 β = 1 p = 2

0.0 0.5 1.0 1.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

x

hrf

a = 1 b = 2 c = 2 k = 2 α = 0.8 β = 0.5 p = 3a = 3 b = 2 c = 2 k = 3 α = 0.2 β = 0.3 p = 1a = 2 b = 1 c = 0.5 k = 2 α = 2 β = 0.5 p = 1a = 1 b = 5 c = 2 k = 0.6 α = 2 β = 2 p = 1.2

Figure 6.3: Plots of (a) density and (b) hrf for KOBD distribution for different parameter

values.

and

f(x) = a b c k α eβ x exp{−α

β

[eβ x − 1

]}(1− exp

{−α

β

[eβ x − 1

]})c−1

(exp

{−α

β [eβ x − 1]})c+1

×{

1 +[exp

{α

β

[eβ x − 1

]}− 1

]c}−k−1

×(

1−{

1 +[exp

{α

β

[eβ x − 1

]}− 1

]c}−k)a−1

×{

1−(

1−{

1 +[exp

{α

β

[eβ x − 1

]}− 1

]c}−k)a}b−1

.

(i) If a = b = 1 in Eq. (6.39), then KOBGo distribution becomes OBGo distribution, (ii) if

a = b = c = 1 in Eq. (6.39), then KOBGo distribution becomes OLxGo distribution, (iii) if

a = b = k = 1 in Eq. (6.39), then KOBGo distribution becomes OLLGo distribution, (iv) If

a = b = c = k = 1 in Eq. (6.39), then KOBGo distribution becomes Gompertz distribution.

Figure 6.4 shows the plots of density and cdf of distribution. The pdf in Figure 6.4 (a)

are right-skewered, left-skewered, symmetrical and bi-model. The hrf in Figure 6.4 (b) are

increasing, bathtub and constant.

105


(a) (b)

0.0 0.5 1.0 1.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

x

pdf

a = 2 b = 0.5 c = 1.5 k = 3 α = 0.5 β = 1.5a = 3 b = 2 c = 1 k = 3 α = 0.8 β = 0.3a = 2 b = 2 c = 0.3 k = 0.6 α = 2 β = 3a = 5 b = 3 c = 0.5 k = 0.5 α = 1 β = 2

0.0 0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

x

hrf

a = 2 b = 0.1 c = 1.5 k = 3 α = 0.5 β = 0.5a = 3 b = 2 c = 1 k = 3 α = 0.2 β = 0.3a = 2 b = 2 c = 0.1 k = 1 α = 2 β = 1.6a = 2 b = 1 c = 0.1 k = 2 α = 2 β = 1.6a = 1 b = 1 c = 1 k = 1 α = 0.3 β = 0.2

Figure 6.4: Plots of (a) density and (b) hrf for KOBGo distribution for different parameter

values.

6.8.5 The Kumaraswamy odd Burr-uniform (KWOBU) distribution.

If Uniform distribution is baseline distribution having pdf and cdf, r(x) = 1/θ and R(x) =

x/θ, then the cdf and pdf of KOBU distribution are, respectively, given by

F (x) = 1−{

1−(

1−{

1 +(

x

θ − x

)c}−k)a}b

. (6.40)

and

f(x) =a b c k

θ

xc−1

(θ − x)c+1

{1 +

(x

θ − x

)c}−k−1(

1−{

1 +(

x

θ − x

)c}−k)a−1

×{

1−(

1−{

1 +(

x

θ − x

)c}−k)a}b−1

. (6.41)

(i) If a = b = 1 in Eq. (6.40), then KOBU distribution becomes OBU distribution, (ii) if

a = b = c = 1 in Eq. (6.40), then KOBU distribution becomes OLxU distribution, (iii) if

a = b = k = 1 in Eq. (6.40), then KOBU distribution becomes OLLU distribution, (iv) if

a = b = c = k = 1 in Eq. (6.40), then KOBU distribution becomes Uniform distribution.

Figure 6.5 shows the plots of density and hrff of KOBU distribution. The pdf in Figure 6.5

(a) are J, reversed-J, symmetrical and bi-model. The hrf in Figure 6.5 (b) are increasing and

bathtub.

106


(a) (b)

0 1 2 3 4

0.0

0.5

1.0

1.5

2.0

x

pdf

a = 4 b = 2 c = 2 k = 2.5 θ = 4a = 0.8 b = 3.5 c = 2 k = 2 θ = 4a = 0.5 b = 0.5 c = 8 k = 0.1 θ = 4a = 0.8 b = 0.8 c = 0.3 k = 4 θ = 4a = 3 b = 3 c = 1.5 k = 0.2 θ = 4

0 1 2 3 4

0.0

0.5

1.0

1.5

x

hrf

a = 0.4 b = 1.5 c = 0.3 k = 3 θ = 4a = 2 b = 3 c = 1.5 k = 0.2 θ = 4a = 0.1 b = 0.8 c = 0.4 k = 0.3 θ = 4a = 0.5 b = 0.8 c = 0.3 k = 4 θ = 4

Figure 6.5: Plots of pdf and hrf for KwOBuU distribution with different parametric values.

The infinite mixture representation of KOBU cdf and pdf in terms of baseline cdf and

pdf are, respectively, given by

F (x) =∞∑

q=0

wq

(x

θ

)q.

f(x) =∞∑

q=0

wq+1q + 1

θ

(x

θ

)q.

The qf of KOBU distribution can be obtained by inverting Eq. (6.40).

Qx(u) = θ

[(1− z)−

1k − 1

] 1c

1 +[(1− z)−

1k − 1

] 1c

,

where z =[1− (1− u)

1b

] 1a .

The rth moment of KOBU distribution can be obtained from Eq. (6.13).

µr =∞∑

q=0

wq+1,q + 1

q + r + 1θr.

The mth incomplete moment of KOBU distribution can be obtained from Eq. (6.14)

T ′s(x) =∞∑

q=0

wq+1q + 1θq+1

xq+s+1

q + s + 1. (6.42)

107


The moment generating function of KOBU distribution can be obtained from Eq. (6.15).

M0(t) =∞∑

q=0

wq+1q + 1θq+1

γ

(q + 1,

−θ

t

)(−1t

)q+1

.

The first incomplete moments of KOBU distribution can be obtained from Eq. (6.42), by

setting m = 1.

T ′1(x) =∞∑

q=0

wq+1q + 1θq+1

xq+2

q + 2(6.43)

The mean deviations of KOBU distribution can be obtained from Eqs. (6.16), (6.16) and

(6.43).

Dµ = 2µF (µ)− 2∞∑

q=0

wq+1q + 1θq+1

µq+2

q + 2.

DM = µ− 2∞∑

q=0

wq+1q + 1θq+1

M q+2

q + 2.

The expression for ith order statistics of KOBU distribution can be obtained from Eq. (6.28).

fi:n(x) =n−i∑

j=0

∞∑

p,q=0

mj(p, q)p + q + 1

θ

(x

θ

)p+q.

The expression for probability weighted moments of KOBU distribution can be obtained

from Eq. (6.30).

E(Xsi:n) = s

n∑

j=n−i+1

(−1)j−n+i−1

j − 1

n− i

n

j

j∑

m=0

∞∑

q=0

j

m

(−1)mem:q

θs

s + q.

6.9 Application

Here, the performance of the KOBG family is accessed by fitting a special model KOBU

and KOBFr distribution on two real data sets.

6.9.1 Data Set 8: Carbon Fibers

The data set has been obtained from Bader and Priest (1982), represents the strength for

the single carbon fibers and impregnated 1000-carbon fiber tows, measured in GPa. It is

reported that the data of single carbon fiber tested at gauge length 1mm.

108


6.9.2 Data Set 9: Birnbaum-Saunders

The data set known by Birnbaum and Saunders (2013) on the fatigue life of 6061-T6 alu-

minium coupons cut parallel to the direction of rolling and oscillated at 18 cycles per sec-

ond is used. The data set contains of 101 values with maximum stress per cycle 31,000 psi.

The KOBFr and KOBU distributions are compared with Beta-Frechet(BFr), Kumaraswamy-

Frechet(KwFr) and Frechet disribtion. R-language is used to estimate the model parame-

ters and model adequacy measures. In Table 6.1 and 6.2 the MLE’s estimates of the param-

eters with associated standard errors with statistics A∗ and W ∗ are given.


Distribution ML estimate W* A*

KOBFr 0.511, 0.288, 49.66, 3.499, 18.97, 0.185 0.033 0.224

S.E 0.241, 0.607, 9.195, 7.135, 7.210, 0.037

KOBU 0.616, 6.09, 7.67, 0.069, 279.2 0.032 0.224

S.E 0.37, 24.85, 3.755, 0.339, 23.42

BFr 54.59, 12.53, 24.35, 0.962 0.216 1.332

S.E 9.70, 5.13, 4.25, 0.207

KwFr 13.07, 103.8, 61.82, 1.26 0.085 0.568

S.E 6.07, 116.2, 22.73, 0.290

Frechet 118.3, 4.27 0.665 4.059

S.E 2.93, 0.267

It is clear from the Tables 6.1 and 6.2, Figures 6.6 and 6.7 that the KOBFr gives better

results as compared to other competitive models.

6.10 Conclusion

In this chapter, a family of distributions called Kumaraswamy Odd Burr XII G family of

distributions is proposed. Some mathematical properties of the family are discussed such

as, rth moment, mth incomplete moment, moment generating function, mean deviations,

109



Distribution ML estimate W* A*

KOBFr 2.55, 6.91, 3.16, 4.57, 4.89, 0.59, 0.025 0.174

S.E 27.62, 83.73, 46.25, 47.59, 60.57, 9.56

KOBU 2.65, 0.59, 2.86, 0.86, 6.16 0.048 0.296

S.E 2.70, 0.83, 1.76, 1.41, 0.28

BFr 0.520, 39.58, 10.52, 1.774 0.028 0.196

S.E 0.565, 47.25, 4.27, 0.875

KwFr 35.17, 61.08, 0.868, 1.304 0.027 0.184

S.E 24.02, 104.1, 0.540, 0.490

Frechet 3.765, 4.420 0.247 1.552

S.E 0.121, 0.397

(a) (b)

x

Den

sity

2 3 4 5 6

0.0

0.1

0.2

0.3

0.4

0.5

KBFrKBLxKBGzKBUBFrKwFrFr

2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

x

cdf

KBFrKBLxKBGzKBUBFrKwFrFr

Figure 6.6: Plots of estimated pdf and cdf for data set 1.

110


(a) (b)

x

Den

sity

100 150 200

0.00

00.

005

0.01

00.

015

0.02

0

KBLxKBFrKBGzKBUBLxKwLxLx

50 100 150 200

0.0

0.2

0.4

0.6

0.8

1.0

x

cdf

KBLxKBFrKBGzKBUBLxKwLxLx

Figure 6.7: Plots of estimated pdf and cdf for data set 2.

entropies, stochastic ordering, moments of residual and reversed residual life, distribu-

tion of ith order statistic and probability weighted moments. Estimation of parameters is

carried out by using the ML method for complete samples and for censored samples. A

special sub model is discussed in detail. Two applications are carried out on real data sets,

to check the performance of the proposed family, which provides consistently better fit

than other competitive models.

111

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