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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
3.5 Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.6 Estimation of parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.6.1 Estimation of parameters in case of complete samples . . . . . . . . . 37
ix
Section 0.0 Chapter 0
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
4.4.1 Stochastic ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.5 Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.6 Maximum Likelihood method . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.7 Properties of OBLP distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.7.1 Simulations study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.8 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
x
Section 0.0 Chapter 0
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
Section 0.0 Chapter 0
6.5.1 Stochastic ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.5.2 Moments of Residual and Reversed residual life . . . . . . . . . . . . 96
6.6 Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.7 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.7.1 Estimation of parameters in case of complete samples . . . . . . . . . 99
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
Section 0.0 Chapter 0
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
parameter values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Plots of the (a) density and (b) hrf for the GBU distribution with different
parameter values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
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
Section 0.0 Chapter 0
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
Section 1.2 Chapter 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
Section 1.2 Chapter 1
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
Section 1.3 Chapter 1
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
Section 1.3 Chapter 1
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
Section 1.3 Chapter 1
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
Section 1.4 Chapter 1
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
Section 1.4 Chapter 1
• 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
Section 2.1 Chapter 2
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
Section 2.1 Chapter 2
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
Section 2.1 Chapter 2
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
Section 2.1 Chapter 2
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
Section 2.2 Chapter 2
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
Section 2.3 Chapter 2
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
Section 2.3 Chapter 2
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
Section 2.4 Chapter 2
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
log[1 + {− log G(xi; ξ)}c
]
+ 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
log[1 + {− log G(xi; ξ)}c
]+ (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
the maximum likelihood estimates.
17
Section 2.4 Chapter 2
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
Section 2.4 Chapter 2
(a) (b)
−4 −2 0 2 4
0.0
0.1
0.2
0.3
0.4
0.5
µ = 0 σ = 1
x
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
Section 2.4 Chapter 2
(a) (b)
0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
1.5
x
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
Section 2.4 Chapter 2
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
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
Section 2.5 Chapter 2
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
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
Section 2.5 Chapter 2
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
Section 2.5 Chapter 2
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
Section 2.5 Chapter 2
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
Section 2.6 Chapter 2
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
Section 2.6 Chapter 2
(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
GBUWUWBXIIBBXIIKwXII
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
Section 2.6 Chapter 2
(a) (b)
x
Den
sity
0 50 100 150
0.00
00.
005
0.01
00.
015
0.02
0
GBUWUWBXIIBBXIIKwXII
0 50 100 150
0.0
0.2
0.4
0.6
0.8
1.0
x
cdf
GBUWUWBXIIBBXIIKwXII
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
Section 3.2 Chapter 3
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 +
[− log R(x; ξ)]c}−k
. (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 +
[− log R(x; ξ)]c}−k
(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
Section 3.2 Chapter 3
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
Section 3.3 Chapter 3
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
Section 3.4 Chapter 3
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
Section 3.4 Chapter 3
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
Section 3.5 Chapter 3
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
likelihood ratio exists between X ≤lr Y .
35
Section 3.6 Chapter 3
3.5 Order Statistics
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)
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 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
Section 3.6 Chapter 3
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.
3.6.1 Estimation of parameters in case of complete samples
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}
The components of score vector are
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
the maximum likelihood estimates.
3.6.2 Estimation of parameters in case of censored complete samples
Suppose that the lifetime of the first r failed items x1, x2, ..., xr have been observed. Then,
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
Section 3.7 Chapter 3
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)
The components of score vector are
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
Section 3.7 Chapter 3
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
Section 3.7 Chapter 3
(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
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
Section 3.7 Chapter 3
(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
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
Section 3.8 Chapter 3
(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
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
are, respectively, given by
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
Section 3.8 Chapter 3
(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
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
Section 3.8 Chapter 3
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
Section 3.8 Chapter 3
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
Section 3.8 Chapter 3
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
Section 3.9 Chapter 3
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
Section 3.9 Chapter 3
(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
Section 3.9 Chapter 3
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
Section 4.1 Chapter 4
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
Section 4.2 Chapter 4
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
Section 4.2 Chapter 4
(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
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
Section 4.2 Chapter 4
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
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
Section 4.2 Chapter 4
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
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
Section 4.3 Chapter 4
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
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
Section 4.3 Chapter 4
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
Section 4.3 Chapter 4
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
Section 4.4 Chapter 4
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
Section 4.4 Chapter 4
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
Section 4.4 Chapter 4
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
Section 4.5 Chapter 4
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).
4.4.1 Stochastic ordering
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
Section 4.6 Chapter 4
4.5 Order Statistics
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)
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 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
Section 4.7 Chapter 4
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}.
The components of score vector are
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
the the maximum likelihood estimates.
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
Section 4.7 Chapter 4
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
Section 4.7 Chapter 4
where zi ={
(1 + xiβ )α − 1
}c.
The components of score vector are
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
Section 4.7 Chapter 4
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
Section 4.7 Chapter 4
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
Section 4.9 Chapter 4
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
Section 4.9 Chapter 4
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
Section 4.9 Chapter 4
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
Section 5.1 Chapter 5
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
are, respectively, given by
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
Section 5.1 Chapter 5
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
Section 5.2 Chapter 5
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
Section 5.2 Chapter 5
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
Section 5.2 Chapter 5
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
Section 5.2 Chapter 5
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]
.
Combining these results in Eq. (5.17), we have
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
Combining these results in Eq. (5.17), we have
ηx = ηT + log(
β2
c k
)2 (β − 1)E (log T ) +
1kE(T β) + (1− c) E (log X) . (5.22)
78
Section 5.3 Chapter 5
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
Section 5.3 Chapter 5
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,
respectively, given by
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
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
Section 5.3 Chapter 5
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
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
Section 5.3 Chapter 5
(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
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
Section 5.4 Chapter 5
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
The components of score vector are
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
the the maximum likelihood estimates.
83
Section 5.4 Chapter 5
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
Section 5.4 Chapter 5
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
Section 5.4 Chapter 5
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
Section 5.5 Chapter 5
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
Section 5.5 Chapter 5
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
Section 6.2 Chapter 6
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,
respectively, given by
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
Section 6.3 Chapter 6
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
Section 6.4 Chapter 6
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
Section 6.4 Chapter 6
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
Section 6.4 Chapter 6
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
Section 6.5 Chapter 6
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.
6.5.1 Stochastic ordering
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
Section 6.6 Chapter 6
From the above expression, we observe that if b1 < b2 ⇒ ddx
f(x)g(x) < 0, then this implies that
likelihood ratio exists between X ≤lr Y .
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
Section 6.6 Chapter 6
6.6 Order Statistics
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)
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
fi:n(x) =1
β(i, n− i + 1)g(x) Gi−1(x) [1−G(x)]n−i
97
Section 6.7 Chapter 6
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
Section 6.7 Chapter 6
6.7.1 Estimation of parameters in case of complete samples
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
the the maximum likelihood estimates.
6.7.2 Estimation of parameters in case of censored complete samples
Suppose that the lifetime of the first r failed items x1, x2, ..., xr have been observed. Then,
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
Section 6.8 Chapter 6
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
the the maximum likelihood estimates.
6.8 Special Sub Models
Here, some special models of KOBG family are considered with their plots of density and
haard rate function.
100
Section 6.8 Chapter 6
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
Section 6.8 Chapter 6
(a) (b)
0 1 2 3 4
0.0
0.5
1.0
1.5
x
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
Section 6.8 Chapter 6
(a) (b)
0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
1.5
2.0
x
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
Section 6.8 Chapter 6
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
Section 6.8 Chapter 6
(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
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
Section 6.8 Chapter 6
(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
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
Section 6.8 Chapter 6
(a) (b)
0 1 2 3 4
0.0
0.5
1.0
1.5
2.0
x
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
Section 6.9 Chapter 6
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
Section 6.10 Chapter 6
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.
Table 6.1: MLEs and their standard errors for data set 1.
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
Section 6.10 Chapter 6
Table 6.2: MLEs and their standard errors for data set 2.
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
Section 6.10 Chapter 6
(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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