Weibull-Lomax Distribution and Its Properties and Applications

Al-Azhar University-Gaza Deanship of Postgraduate Studies Faculty of Economics and Administrative Sciences Department of Statistics

Weibull-Lomax Distribution and its Properties and Applications

توزيع ويبل - لومكس و خصائصه و تطبيقاته

By: Ahmed Majed Hamad El-deeb

Supervisor: Prof. Dr. Mahmoud Khalid Okasha

A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master in Statistics

Gaza, 2015

Abstract

The Weibull distribution is a very popular probability mathematical distribution for its flexibility in modeling lifetime data particularly for phenomenon with monotone failure rates. When modeling monotone hazard rates, the Weibull distribution may be an initial choice because of its negatively and positively skewed density shapes. However, it does not provide a reasonable parametric fit for modeling phenomenon with non-monotone failure rates such as the bathtub and upside-down bathtub shapes and the unimodal failure rates which are common in reliability, biological studies and survival studies.

In this thesis we introduce and discuss a new generalization of Weibull distribution called the Four-parameter Weibull-Lomax distribution in an attempt to overcome the problem of not being able to fit the phenomenon with non-monotone failure rates. The new distribution is quite flexible for analyzing various shapes of lifetime data and has an increasing, decreasing, constant, bathtub or upside down bathtub shaped hazard rate function. Some basic mathematical functions associated with the proposed distribution are obtained.

The Weibull-Lomax density function could be wrought as a double mixture of Exponentiated Lomax densities. And includes as special cases the exponential, Weibull. Shapes of the density and the hazard rate function are discussed. The four parameters of the Weibull-Lomax distribution were estimated using the maximum likelihood approach. The Truncated Weibull-Lomax distribution was introduced with two special cases of the truncation. Basic mathematical functions associated with the Truncated Weibull-Lomax distribution are obtained. The results are illustrated using two real datasets on lifetime of breast cancer patients in Gaza Strip and failure time of Aircraft Windshield. The Weibull-Lomax distribution has been fitted together with other well-Known distributions in modeling lifetime data and it has been proved to provide the best fit among all other distributions with respect to seven well-known goodness–of–fit criteria. However, the Truncated Weibull- Lomax distribution provides a better fit than the original one.

الملخص

يعتبر توزيع ويبل أحد التوزيعات الرياضية األكثر استخداماً في نمذجة بيانات الحياة ، و لنمذجة الظواىر ذات معدل مخاطرة مطرد، يعتبر توزيع ويبل الخيار األساسي وذلك بسبب أن شكل دالة كثافة

االحتمال ليذا التوزيع ينحرف يمنةً أو يسرةً . إال أنو وبالرغم من كل ذلك فإنو ال يممك األسباب الكافية لنمذجة الظواىر الغير مطردة مثل الظواىر ذات الشكل األشبو بحوض االستحمام وحوض االستحمام

المقموب، التي تعتبر شائعةً في كل من د ارسة الموثوقية والبقاء. لذلك يعتبر ويبل من التوزيعات األكثر مرونة في نمذجة بيانات الحياة ذات الطابع المطرد في معدل المخاطرة ولكنو غير عممي في نمذجة الظواىر الغير

مطردة.

في ىذه الدراسة نناقش تعميم جديد لتوزيع ويبل وىو توزيع ويبل لومكس ذو األربع معالم في محاولة

لمتغمب عمى مشكمة عدم مقدرة توزيع ويبل عمى تقدير الظواىر ذات معدل مخاطرة غير مطردة. ويعتبر

توزيع ويبل لومكس توزيعاً مرناً بقدر كافي لتحميل بيانات الحياة لظواىر ذات معدل مخاطرة ذو شكل ت ازيدي، تناقصي، ثابت، شبيو بحوض االستحمام أو حوض االستحمام المقموب، وتمت دراسة بعض الخواص

والمعادالت الرياضية المرتبطة بالتوزيع الجديد.

وقد تمكنا من اظيار أن توزيع ويبل لومكس يحتوي حاالت خاصة وىي التوزيع األسي و توزيع

ويبل . كما تمكنا من صياغة دالة كثافة االحتمال لتوزيع ويبل لومكس كتوزيع مركب من كثافة احتمال توزيع

لومكس األسي ، وتمت مناقشة شكل كل من دالة كثافة االحتمال ومعدل المخاطرة لمتوزيع الجديد، وقدرنا

المعالم األربعة لو بواسطة دالة األرجحية العظمى، عالوة عمى ذلك، قمنا بتقديم ومناقشة توزيع ويبل لومكس

المبتور بحالتيو الخاصتين لجية البتر، ودراسة بعض الخواص والمعادالت الرياضية المرتبطة بتوزيع ويبل

لومكس المبتور. وفي النياية، تم تقدير معالم توزيع ويبل لومكس ومجموعة مختمفة من التوزيعات الرياضية

المعروفة في نمذجة بيانات الحياة لنمذجة مجموعتين من البيانات أحدىما لبيانات فترة الحياة لمرضى سرطان

الثدي في قطاع غزة، واألخرى لفترة بقاء الزجاج األمامي لمطائرات. وأظيرت النتائج بأن توزيع ويبل لومكس

كان أفضل نموذج في ظل النماذج األخرى المقارنة معو وذلك بنا ًء عمى سبعة من معايير جودة التوفيق االحصائية في كال من مجموعتي البيانات. وعند تطبيق توزيع ويبل لومكس المبتور بدا أنو أفضل من توزيع

ويبل لومكس األصمي.

Acknowledgement

Firstly, I would like to express my sincere gratitude to my advisor Prof. Dr. Mahmoud Okasha for the continuous support of my Master study and related research, for his patience, motivation, and immense knowledge. His guidance helped me all the time of during research and writing of this thesis. I could not have imagined having a better advisor and mentor for my Master research.

Besides my advisor, I would like to thank the rest of my thesis committee: Prof. Dr. Abdullah Elhabeel, department of Statistics at Al-Azhar University and Dr. Raed Salha, department of Mathematics at Islamic University, for accepting to evaluate my search.

Last but not the least, I would like to thank my family: my father Dr. Majed El-deeb and my mother and my brothers and sister for supporting me spiritually throughout writing this thesis and my life in general.

III

TABLE OF CONTENTS

Subject Page No.

List of Tables …………………………………………...……………... X

List of Figures …...…………………………………………………….. XI

Abbreviations …………………………………………………………. XIII

Chapter 1. Introduction …………………………………………. 1

1.1. Rationale ………………………………………………..…… 1

1.2. Lifetime Modeling ……………………………………..……. 3

1.3. Research problem ………………………………………....… 5

1.4. Motivation ……………………………………………..….… 6

1.5. Research Objectives ……………………………………....… 6

1.6. Research Methodology ………………………………...….… 7

1.7. Research Importance ………………………………...……… 7

1.8. Literature Review …….…………………………….………... 8

1.9. Organization of This Thesis ………………………..…...…... 10

Chapter 2. Development of Lifetime Distributions ………….. 12

2.1. Introduction …………………………………………..…..…. 12

2.2. The Weibull distribution ……………………...………...…… 14

2.3. The Lomax distribution ………...……………………..…….. 18

2.4. Mixture Representation of the Density Function of the Weibull Distribution ……………………………………..………. 20

2.5. Weibull-Pareto distribution ………………………...……...... 22

Chapter 3. The four-parameter Weibull-Lomax distribution ... 24

3.1. Introduction ……………………………...………………….. 24

3.2. The Weibull-Lomax distribution ………………………...….. 25

3.2.1. The Cumulative distribution function of the …………………….…………………..………… 26

3.2.2. The probability density function of the ….. 28

3.3. Transformations on the Weibull-Lomax distribution ………. 30

3.4. Annotation of the Weibull-Lomax distribution: Mixture representation of the WL pdf ...... 32

3.5. Mathematical structural properties of Weibull Lomax Distribution ………...………………………………………... 35

3.5.1. The Hazard rate function and other related functions of the truncated Weibull Lomax ( ) distribution …...... 35

3.5.1.1. The Survival function of the Weibull Lomax ( ) distribution ……………………..……………...….. 35

3.5.1.2. The Hazard rate function of the Weibull Lomax ( ) distribution ………..…………………………...….. 36

3.5.1.3. The reversed-hazard rate function (rhrf) function of the Weibull Lomax ( ) distribution ……………....………. 36

3.5.1.4. The cumulative hazard rate function function of the Weibull Lomax ( ) distribution ……………..…. 38

3.5.2. Quantile distribution function ……………………...……. 38

3.5.3. Moments of the Weibull-Lomax distribution ………….... 40

3.5.3.1. The Mean of the WL distribution ………………..….. 42

3.5.3.2. The second moment of the WL distribution ……….... 42

3.5.3.3. The third moment of the WL distribution ………..…. 43

3.5.3.4. The Forth moment of the WL distribution …….…..... 44

3.5.3.5. The Variance of the Weibull Lomax distribution ….... 46

3.5.3.6. The Skewness of the Weibull Lomax distribution …… 47

3.5.3.7. The Kurtosis of the Weibull Lomax distribution …..... 47

3.5.4. Coefficient of Variation ………………………………..… 48

3.5.5. Incomplete moments ………………...…………………... 48

3.5.6. Bonferroni and Lorenz Curves …………………...……… 50

3.5.7. The moment generating function ……………..…………. 51

3.5.8. Mean deviations ……………………..…………………... 52

3.5.9. Mean residual life ……………………..………………… 53

3.5.10. Entropies …………….…………..……………………... 53

3.5.11. The Mode of the WL ( ) distribution ...……. 54

3.6. Shapes of the density and hazard rate functions …...………... 56

3.6.1. The limits of the density function and the hazard rate function ……...…………………………………………………. 56

3.6.2. Shapes of the hazard rate function ………...…………….. 57

3.7. Summary ………...…………………………………………... 60

Chapter 4. Estimation of the parameters of the Weibull-Lomax distribution ………………………………………………………. 62

4.1. Introduction ………...……………………………………….. 62

4.2. Maximum Likelihood Estimators of the parameters of the Weibull-Lomax distribution …………………………………...… 62

4.3. Confidence interval estimates of the parameters …………… 65

4.4. Simulation Study …...……………………………………….. 67

4.5. Summary ………...…………………………………………. 70

Chapter 5. Inference on the four-parameter truncated Weibull-Lomax distribution …….……………………………… 71

5.1. Introduction ………………...……………………………..… 71

5.2. The distribution function of the four-parameter T-WL distribution …………………………………………………..…… 72

5.2.1. The probability density function of the doubly truncated ……..……………………………………….…... 72

5.2.2. The Cumulative distribution function of the doubly truncated …………………...……………..…….. 74

5.3. Special cases the truncated Weibull Lomax ( ) ……………………………..……………………………… 78

5.3.1. A special case: The left truncated Weibull Lomax ( ) …………………………………………...………... 78

5.3.2. A special case: The right truncated Weibull Lomax ( ) ………………………...………………………….. 78

5.4. Mathematical properties of the truncated Weibull Lomax distribution ……………………………………………..………… 79

5.4.1. The Hazard rate function and other related functions of the truncated Weibull Lomax ( ) distribution …….…..... 79

5.4.1.1. The Survival function of the truncated Weibull Lomax ( ) distribution ……………...………..….. 79

5.4.1.2. The Hazard rate function of the truncated Weibull Lomax ( ) distribution ………...……...… 80

5.4.1.3. The reversed-hazard rate function (rhrf) function of the truncated Weibull Lomax ( ) distribution ……….. 80

5.4.2. The moments of the truncated Weibull Lomax distribution ……………..…………………………….…… 81

5.4.2.1 The Mean of the truncated Weibull Lomax distribution .…………………….………………………….….. 82

5.4.2.2. The second moment of the truncated Weibull Lomax distribution ……………………………………………...….…. 83

5.4.2.3. The third moment of the truncated Weibull Lomax distribution ……………………….………..…………………. 84

5.4.2.4. The Forth moment of the truncated Weibull Lomax distribution …………………………………………...…….… 85

5.4.2.5. The Variance of the truncated Weibull Lomax distribution …………………………………………………… 85

5.4.2.6. The Skewness of the truncated Weibull Lomax distribution ………………………..………………………….. 87

VII

5.4.2.7. The Kurtosis of the truncated Weibull Lomax distribution …………………………………..………………. 87

5.4.3. Coefficient of Variation ……………………………….... 88

5.4.4. The Mode of the truncated Weibull Lomax ( ) distribution ……………………………………………………… 88

5.4.5. Incomplete moments ……………………………….…..... 90

5.4.6. Bonferroni and Lorenz Curves …………...………...……. 92

5.4.7. Moment Generating Function ………………………....…. 93

5.4.8. Mean deviations …………………………………..……… 94

5.4.9. Mean residual life ………………………………..……..... 95

5.4.10. The Quintile function of the truncated Weibull Lomax( ) distribution ……………………...………. 96

5.5. Summary …………………………………………………..… 98

Chapter 6. Application of the Weibull-Lomax and Truncated Weibull-Lomax distributions to two real data sets ……………. 99

6.1. Introduction ………………………………………………..... 99

6.2. Possible alternative distributions …………………………..... 99

6.3. The Data and the distributions' fits …………...……………... 101

6.3.1. First data set: Lifetime of Breast Cancer Patients' Data in Gaza Strip ……………………………………………..………… 101

6.3.1.1. Estimation of the parameters of Weibull Lomax ( ) distribution from the lifetime breast cancer patients' data in Gaza Strip ………………………………….…………... 105

6.3.1.2. Expected lifetime of breast cancer patients from original lifetime data …………………………………………..……...... 105

6.3.2. The second data set: Failure times of Aircraft Windshield ……………………………………………………… 105

6.3.2.1. Estimation of the parameters of Weibull Lomax ( ) distribution from the failure times of Aircraft Windshield data …………....…………………………………... 107

VIII

6.3.2.2. Expected mean lifetime of Aircraft Windshield from original lifetime data …….………………………………...…… 109

6.4. The truncated Data and the distributions' fit ………………... 109

6.4.1. First data set: Lifetime of Truncated Breast Cancer Patients' Data in Gaza Strip ………………...…………………… 109

6.4.1.1. Estimation of the parameters of Truncated Weibull Lomax ( ) distribution from the truncated lifetime breast cancer patients' data ………………………………………..….. 111

6.4.1.2. Expected mean lifetime of breast cancer patients from the truncated lifetime data ……………………………….…….. 111

6.4.2. The second data set: Failure times of (67) Truncated Aircraft Windshield ……………………………………….…..... 113

6.4.2.1. Estimation of the parameters of Truncated Weibull Lomax ( ) distribution from the failure times of Truncated Aircraft Windshield data ………………………….... 114

6.4.2.2. Expected mean lifetime of Aircraft Windshield from truncated lifetime data ………………………………………..... 116

6.5. Summary …………………………………………..……….... 116

Chapter 7. Conclusion and Recommendations ………………... 117

7.1. Introduction ………………………………...……………..… 117

7.2. Conclusions ……………………………………………….… 117

7.3. Recommendations …………………………………………... 119

References ……………………………………………………….. 120

List of Tables

Table Page No.

Table 4.1. Estimated MLEs, SEs of the MLEs of parameters of WL distribution based on 1000 simulations with n=25, 50, 200, 500 and 1000. …………………………………………….. 86

Table 6.1. Data of lifetime (days) of (242) breast cancer patients in Gaza Strip. ………………………………………………... 102

Table 6.2. MLEs of the parameters and its SE's of different distributions for breast cancer patients in Gaza Strip data set. ………………………………………………………… 103

Table 6.3. Goodness of fit of different distributions for breast cancer patients in Gaza Strip data set. ……………………………. 103

Table 6.4. Data of failure times of (84) Aircraft Windshield. ……….. 106

Table 6.5. MLEs and its SE's of collection of different distributions for Aircraft Windshield data set. ………………………... 106

Table 6.6. Goodness of fit of collection of different distributions for the Aircraft Windshield set. ……………………………… 107

Table 6.7. Truncated lifetime data (in days) of (175) breast cancer patients in Gaza Strip. …………………………………….. 110

Table 6.8. MLEs of the parameters and its SE's of different distributions for the truncated breast cancer patients in Gaza Strip data set. ……………………………………….. 110

Table 6.9. Goodness of fit of different distributions for the truncated breast cancer patients in Gaza Strip data set. …………….. 110

Table 6.10. Data of failure times of (67) Truncated Aircraft Windshield. ……………………………………………….. 113

Table 6.11. MLEs and its SE's of collection of different distributions for Truncated Aircraft Windshield data set. ……………... 113

Table 6.12. Goodness of fit of collection of different distributions for the truncated Aircraft Windshield set. ……………………. 114

List of Figures

Figure Page No.

Fig. 2.1. The bathtub curve: the three stages over the device lifetime: high initial failure, constant failure rate over the useful lifetime, and increased failure rate as the devices age. …….... 13

Fig. 2.2. The pdf of Weibull function for fixed value of α and various 15 value of β=0.5 ,1 ,2 ,4. …………………………..…..………..

Fig. 2.3. The cdf of Weibull Function for fixed value of α and various 15 value of β=0.5 ,1 ,2 ,4. …………………….……………..…...

Fig. 2.4. Hazard rate for the Weibull function for fexid value of and various value of . ………………………... 16

Fig. 2.5. The Weibull Data Plot: Time-to-Failure. …………………..... 17

Fig. 3.1.(a) Plots of the WL cdf for and various value of . …………………………………………………….... 27

Fig. 3.1.(b) Plots of the WL cdf for and various value of . ………………………………………………………... 27

Fig. 3.2.(a) Plots of the WL pdf for and various value of . ……………………………….…………………… 29

Fig. 3.2.(b) Plots of the WL pdf for and various value of . ……………………………………………………. 29

Fig. 3.3.(a) Bathtub curves of Hazard rate function for and various value of . ………………….. 37

Fig. 3.3.(b) Upside-down Bathtub curves of Hazard rate function for and various value of . …... 37

Fig. 3.3.(c) Increasing curves of Hazard rate function for and various value of . …………………..... 37

Fig. 3.3.(d) decreasing curves of Hazard rate function for and various value of . ………………… 37

Fig. 3.3.(e) Conestant curves of Hazard rate function for and various value of . …………………..... 37

Fig. 5.1. Different density functions of the truncated WL ( ) with fixed value of left truncation point at and various values of right truncation point at . ………………. 73 Fig. 5.2. Different density functions of the truncated WL ( ) with various value of left truncation point at and fixed value of right truncation point at . …………………………. 74

Fig. 5.3. Cumulative distribution functions of the truncated WL ( ) with fixed value of left truncation point at and various values of right truncation point at . …………..…… 77

Fig. 6.1.(a) Histogram of breast cancer patients in Gaza Strip data set and its fitted of pdfs for WL, ExpdLx, ELP, Lomax. ………….. 104

Fig. 6.1.(b) Histogram of breast cancer patients in Gaza Strip data set and its fitted of pdfs for WL, KwGLx, GuLx, Weibull. ………… 104

Fig. 6.2.(a) Histogram of Aircraft Windshield data set and its fitted of pdfs for WL, ExpdLx, ELP, Lomax. ……………………..... 108

Fig. 6.2.(b) Histogram of Aircraft Windshield data set and its fitted of pdfs for WL, KwGLx, GuLx, ExplLx. ………………………. 108

Fig. 6.3.(a) Histogram of truncated breast cancer patients in Gaza Strip data set and its fitted of pdfs for T-WL, WL, ExpdLx, ELP, Lomax. ………………………………………………………... 112

Fig. 6.3.(b) Histogram of truncated breast cancer patients in Gaza Strip data set and its fitted of pdfs for T-WL, KwGLx, GuLx, Weibull. ……………………………………………………… 112

Fig. 6.4.(a) Histogram of truncated Aircraft Windshield data set and its fitted of pdfs for T-WL, WL, ExpdLx, ELP, Lomax. ………………………………………………………….…….... 115

Fig. 6.4.(b) Histogram of truncated Aircraft Windshield data set and its fitted of pdfs for T-WL, KwGLx, GuLx, ExplLx. ………………………………………………………….………. 115

XII

Abbreviations

Full Word Abbreviation

Anderson-Darling. ………………………………………………….

Akaike information criterion. ……………………………………… AIC

Bayesian information criterion. ……………………………………. BIC

Consistent Akaike information criteria. …………………………… CAIC

Coefficient of Variation. …………………………………………... CV

Cumulative Distribution Function. ……………………………….. cdf

Cumulative Hazard Rate Function. ………………………………... chrf

Exponential Lomax Distribution. ………………………………….. ExplLx

Exponentiated Lomax Poisson Distribution. ………………………. ELP

Exponentiated Lomax Distribution. ……………………………….. ExpdLx

Hannan–Quinn information criterion. ……………………………... HQIC

Hazard Rate Function. ……………………………………………... hrf

Gumbel-Lomax Distribution. ……………………………………… GuLx

Kumaraswamy-Generalized Lomax Distribution. ………………… KwGlx

Maximum Likelihood Estimation. ………………………………… MLE

Moment Generating Function. …………………………………….. mgf

Mean Residual Life Function. ……………………………………... MRL

Probability Density Function. ……………………………………... pdf

Reversed-Hazard Rate Function. ………………………………….. rhrf

Survival Function. …………………………………………………. sf

Truncated Weibull Lomax Distribution. …………………………... T-WL

Weibull Lomax Distribution. ……………………………………… WL

Cramér-von Mises statistic. ………………………………………...

XIII

Chapter 1

Introduction

1.1. Rationale

The concept of reliability was initially introduced in the beginning of the twentieth century. Since then, a great number of research and applications have been carried out in order to explore and understand the methodologies and applications of reliability analysis for product enhancements. Reliability is generally regarded as the likelihood that a product or service is functional during a certain period of time under a specified operational environment. Reliability is always considered as one of the most important characteristics for industrial products and systems. Reliability engineering studies the lifetime data and subsequently uses it to estimate, evaluate and control the capability of components, products and systems. The theories and tools of reliability engineering is applied into widespread fields such as electronic and manufacturing products, aerospace equipments, earthquake and volcano forecasting, communication systems, navigation and transportation control, medical treatment to the survival analysis of human being or biological species and so on (Weibull, 1977; Lawless, 1982).

With the increased complexity of component structure and the continuous requirements of high quality and reliability products, the role of reliability of the product became more important to both the producers and the consumers nowadays. It is believed that unreliable components or systems will cause inconvenience to the productivity in our daily lives. In even worse situations, any unstable component of a product can cause huge economic loss and serious damage to customers, producers, government and the society.

For instance, the recent cracking of the United States space shuttle Columbia on February 1, 2003 caused the death of all the seven astronauts on board. Such disaster was investigated and announced later that the worn-out and obsolete state of the 20 year old space shuttle might account for the accident. The investigation committee also pointed out that early preventive measure should have been taken for the outside material of the shuttle. It is clear that the risk of cracking will be better controlled as along as more accurate reliability testing and estimation and regular maintenance work are carried out in time.

In 1986, the failure of the sealing material of another US booster rocket on space shuttle Challenger directly resulted in the explosion of the whole space shuttle. The reliability of the sealing material in one of the system equipment was vital to the stable usage and running of the shuttle. If the shuttle designers had a better understanding of the nature or reliability of the sealing component under severe environments, such a disaster exposure on the whole system can be avoided. Reliability engineers must fully involve and utilize their reliability knowledge in the stage of designing, testing and maintaining of the shuttle.

Another medical accident happened in the late of 1980s in Salt Lake City, where approximately 150 heart disease patients died due to the unreliable mechanical human hearts replaced after their medical operations. The functionalities of the product were not fully verified and precisely predicted before the patients went through those medical operations. When doctors encountered difficulties in planning and conducting medical trials, an integrated testing process and accurate estimate of the functionality of the medical products before their introduction to the practice were critical to the success of the medical treatment and survival of the patients.

From these examples, it can be concluded that high reliability is strictly required for the functionality of the systems and safety of people using the products. The increased emphasis on reliability is also due to considerable other

2 factors, including awareness of stability of high quality products, complexity and sophistication of systems, new industrial regulations concerning product liability, government contractual requirements on performance specifications and product cost for testing, repairing and warranty (Kececioglu, 1991 and Ebeling, 1997). Consequently, billions of dollars are invested into the research of reliability engineering to improve the stabilities of the products during the decades. It is therefore meaningful for us to continuously investigate research related to reliability modeling (Tang, 2004).

1.2. Lifetime Modeling

After collecting lifetime data for the analysis, it is apparent that a suitable and valid reliability model is essential to the feasibility of model estimation and analysis. As a result, various reliability models have been generated for data analysis; for example, the frequently used Exponential distribution, Weibull distribution, Normal distribution, Lognormal distribution, Gamma distribution and so on. Before the 1980s, most products were assumed to follow exponential distribution, which has the simplest mathematical form with tractable statistical properties. Products following an exponential lifetime distribution have the so- called no-memory property. However, it is found out later that the assumptions of the exponential distribution must always be taken into consideration in order to have more accurate predictions of the underlying failure mechanism. Hence, different models should be utilized under complex situations when the assumption of constant random failure rate is restrictive (Tang, 2004).

Among these statistical models, Weibull system of distributions was constructed in 1939, Weibull distribution named after the Swedish Professor Waladdi Weibull. Weibull (1939) introduced form of cumulative distribution function for modeling lifetime data. Weibull distribution, is perhaps the most frequently used life time distribution for lifetime data analysis mainly because of not only its flexibility of analyzing diverse types of aging phenomena, but also its simple and straightforward mathematical forms compared with other distributions.

Several authors consider different aspects of the Weibull distribution; see for example Thoman et al. (1969); Scholz (1996); Rinne (2009); Weibull (1951); Murthy et al. (2003) and Nichols & Padgett (2006).

The attractiveness of this relatively known distribution for model fitting is that it combines a simple mathematical expression for the cumulative frequency function and probability density function with wide coverage in the skewness and kurtosis plane.

The Weibull distribution is a very popular model and has been extensively used over the past decades for modeling data in reliability, engineering and biological studies. It is generally adequate for modeling monotone hazard rates but it is not useful for modeling the bathtub shaped and the unimodal failure rates which are common in survival, reliability and biological studies. Thus it can't be used to model lifetime data with a bathtub shaped hazard function, such as human mortality and machine life cycles (Mudholkar & Hutson, 1996).

Numerous classical distributions have been extensively used over the past decades for modeling data in several areas such as engineering, actuarial, environmental and medical sciences, biological studies, demography, economics, finance and insurance. However, in many applied areas such as lifetime analysis, finance and insurance, there is a clear need for extended forms for these distributions. For that reason, several methods for generating new families of distributions have been studied.

Some attempts have been made to define new families of probability distributions that extend well-known families of distributions and at the same time provide great flexibility in modeling data in practice. One such example is a broad family of univariate distributions generated from the Weibull distribution introduced by Gurvich et al. (1997), by extending the classical Weibull model (Bourguignon et. al., 2014).

In this thesis, we propose and study an extension of the Weibull and Lomax models and at the same time provide great flexibility in modeling data in practice. called the Weibull-Lomax (“WL” for short) distribution.

1.3. Research problem

In practice, the Weibull distribution, having exponential and Rayleigh as special sub-models, is a very popular distribution for modeling lifetime data and for modeling phenomenon with monotone failure rates. When modeling monotone hazard rates, the Weibull distribution may be an initial choice because of its negatively and positively skewed density shapes. However, it does not provide a reasonable parametric fit for modeling phenomenon with non-monotone failure rates such as the bathtub and upside-down bathtub shapes and the unimodal failure rates that are common in reliability and biological studies. Such bathtub hazard curves have nearly flat middle portions and the corresponding densities have a positive anti-mode. An example of bathtub shaped failure rate is the human mortality experience with a high infant mortality rate which reduces rapidly. It then remains at that level for quite a few years before picking up again. Unimodal failure rates can be observed in course of a disease whose mortality reaches a peak after some finite period and then declines gradually. Recently several generalization of Weibull distribution has been studied. An approach to the construction of flexible parametric models is to embed appropriate competing models into a larger model by adding shape parameter. Some recent generalizations of Weibull distribution including the Exponentiated Weibull, extended Weibull, modified Weibull are discussed in Pham & Lai (2007) and references therein, along with their reliability functions (Aryal & Tsokos, 2011).

Thus, the Weibull distribution has been shown to be very flexible in modeling various type of lifetime distribution with monotone failure rates but it is not useful for modeling the bathtub shaped and the unimodal failure rates which are common in reliability, biological studies and survival studies such as human mortality and machine life cycles.

In this thesis we discuss a new generalization of Weibull distribution called the Weibull-Lomax distribution. We will derive the subject distribution using the Weibull-G Family studied by Bourguignon et al.(2014).

1.4. Motivation

We are motivated to introduce this distribution because of the wide usage of the general class of Weibull distributions. The structure of the density function of the new distribution can be motivated as it provides more flexible distribution than the standard two-parameter Weibull and many other generalized Weibull distributions.

1.5. Research Objectives

The main objectives of this thesis are to:

 Propose a new generalization to each of Weibull and Lomax distributions called Weibull-Lomax (“WL” for short) distribution, and investigate various properties for Weibull-Lomax distribution .

 Use the maximum likelihood procedures to derive such estimates, give a confidence interval for estimate parameters, make a simulation study, and use it to estimate the parameters and standard error.

 Propose a new generalization to the Weibull-Lomax called doubly truncated Weibull-Lomax distribution with four parameters, investigate various properties for truncated Weibull-Lomax distribution.

 Examination the flexibility of the two proposed distributions by goodness of fit and some illustration plots by employment of two data sets by using R package.

1.6. Research Methodology

To achieve the previous goals the following methods and techniques will be used:

 Statistical theories such as expectation, variance, moments generating functions and some probability distribution functions such as Exponentiated Lomax, Weibull, Lomax and Beta distributions, Weibull-G family, complete and incomplete beta functions and Truncated theoretical methodology, among others.  Estimation theories such as maximum likelihood estimators (MLE) will be used.  Theoretical results will be illustrated using simulation methods and applied on two real datasets of Breast cancer patients in Gaza Strip and failure times of Aircraft windshield.  The R Software will be used in this thesis.

1.7. Research Importance

In some instances physical considerations determine the distribution to be employed. However, numerous statistical problems arise from physical processes which cannot be theoretically linked to a particular parametric distribution. The lack of a general theory for the choice of an underlying parametric distribution has resulted in the usage of some distributions which are not necessarily robust to viable alternative distributions and which may lead to incorrect conclusions.

The emphasis of this research is placed upon the Creation of an underlying parametric distribution that can be used in survival analyses or reliability studies. Since the reliability of a piece of equipment can be modeled by a probability statement concerning its lifetime operation, there is a close connection between reliability theory and survival theory, where it is desired to make probability statements about survival or remission times of patients.

Weibull-Lomax distribution it seems to be an important distribution that can be used in a variety of problems in modeling survival data, if it contains sufficient flexibility in accommodating different forms of the risk function.

Also Weibull-Lomax distribution it seems to be an important distribution that can be used in a variety of problems in modeling survival data, if it is not only convenient for modeling comfortable bathtub-shaped failure rates but it is also suitable for testing the goodness-of-fit of some distributions.

1.8. Literature Review

The Weibull distribution is a popular lifetime distribution model in reliability engineering. However, this distribution does not have a bathtub or upside–down bathtub shaped hazard rate function, which is why it cannot be utilized to model the lifetime of certain systems. To overcome this shortcoming, several generalizations of the classical Weibull distribution have been discussed by different authors in recent years. Many authors introduced flexible distributions for modeling complex data and obtaining a better fit. Many extended Weibull models have a bathtub and an upside-down bathtub shaped hazard rate. Extensions of the Weibull distribution arise in different areas of research as discussed for instance in,

Ghitany et al. (2005) who introduced a two-parameter Marshall-Olkin extended Weibull distribution which can be obtained as a compound distribution with mixing exponential distribution. Nichol and Padgett (2006) developed a parametric bootstrap method in establishing the lower and upper control limits for monitoring percentiles when process measurements have a Weibull distribution. Carrasco et al.

(2008) proposed a four-parameter generalized modified Weibull distribution. For the same motivations, Zografos and Balakrishnan (2009) proposed the beta- and generalized gamma-generated distributions. Barreto et al. (2010) generalized a three-parameter Weibull-Geometric distribution. Cordeiro et al. (2010) developed the four-parameter Kumaraswamy Weibull distribution, Silva et al. (2010) proposed the five-parameter beta modified Weibull distribution. Aryal and Tsokos (2011) introduced a three-parameter transmuted Weibull distribution. Provost et al.

(2011) generalized the three-parameter gamma-Weibull distribution. Pinho et al.

(2012) introduced a four-parameter gamma-exponentiated Weibull distribution. Singla et al. (2012) generalized the five-parameter Beta - Generalized Weibull (BGW) distribution. Badmus & Ikegwu (2013) introduced the five-parameter Beta Weighted Weibull Distribution. Cordeiro et al. (2013a) proposed the three- parameter Exponential-Weibull lifetime distribution. Cordeiro et al. (2013b) proposed a new method of adding two parameters to a continuous distribution named by the Exponentiated generalized distributions. Cordeiro et al. (2013c) introduced the five-parameter beta Exponentiated Weibull distribution. Cordeiro et al. (2013d) introduced the five-parameter beta–Weibull geometric distribution. Cordeiro and Lemonte (2013) developed the Marshall-Olkin extended Weibull family MOEW (δ, α, ξ ), where ξ is a parameter vector of a baseline distribution. Cordeiro et al. (2014a) introduced a five-parameter McDonald - Weibull distribution. Cordeiro et al. (2014b) modified the five-parameter Kumaraswamy- Weibull distribution and discussed its theory and applications. Peng and Yan (2014) introduced the new three-parameter extended Weibull distribution. Saboor el al. (2014a) introduced a six parameter generalized extended inverse Gaussian.

Saboor and Pog´any (2014) developed the four-parameter Marshall–Olkin gamma– Weibull Distribution with Applications. Tojeiro el al. (2014) introduced the three- parameter complementary Weibull geometric distribution. Jiang and Murthy

(1998) developed a Mixture of Weibull distributions–parametric characterization of failure rate function. Finally, Nadarajah el al. (2011) proposed the four- parameter the beta-modified Weibull distribution.

Adding new shape parameters to expand a model into a larger family of distributions to provide significantly skewed and heavy-tails plays a fundamental role in distribution theory. More recently, there has been an increased interest in defining new univariate continuous distributions by introducing additional shape parameters to the baseline model. There has been an increased interest in defining new generators for univariate continuous families of distributions by introducing one or more additional shape parameter(s) to the baseline distribution. This

9 induction of parameter(s) has been proved useful in exploring tail properties and also for improving the goodness-of-fit of the proposed generator family. This usual distribution does not have a bathtub hrf. However, several distributions derived from the basic Weibull distribution have bathtub-shaped hrf. Two of these are the exponentiated Weibull distribution proposed by Mudholkar & Srivastava, (1993), and the competing risk and sectional models involving two Weibull distributions, discussed in Jiang & Murthy (1995) and Lai et al. (2003).

1.9. Organization of this Thesis

The present thesis has the following set up:

Chapter 2, contains detailed discussion on inference about the development of lifetime distributions with different parameters, and it is divided into five main parts:  The distribution function of the Weibull ( ) distribution.  The distribution function of the Lomax ( ) distribution.  Mixture Representation of the Density Function of the Weibull Distribution.  Weibull-Pareto distribution as one of many generalizations of the Weibull distribution.

In Chapter 3, we introduce the new distribution named "Weibull-Lomax distribution" and provide some detailed discussion on inference on the distribution with four parameters. This chapter is divided into six major sections:  Defined our new model "Weibull-Lomax distribution" with four parameters.  Transformations on the Weibull-Lomax distribution.  Annotation of the Weibull-Lomax distribution: Mixture representation of the Weibull-Lomax probability density function.  Mathematical and structural properties of Weibull Lomax distribution.  Shapes of the density and hazard rate functions.

Chapter 4, contains estimation to the four parameters of the Weibull- Lomax distribution, building of a confidence interval for the four parameters. This chapter is divided into three major sections:  Maximum Likelihood Estimators of the parameters of the Weibull-Lomax distribution.  Confidence interval estimates of the parameters.  Simulation Study.

In Chapter 5, we discuss in details, inference on the truncated Weibull- Lomax distribution with four parameters and provide some detailed discussion on inference on the new truncated distribution. This chapter is divided into three major sections:  The distribution function of the four-parameter T-WL distribution.  Special cases of the truncated Weibull Lomax ( ) distribution.  Mathematical and Structural properties of the truncated Weibull Lomax( ) distribution.

In Chapter 6, we will apply Weibull Lomax distribution and truncated Weibull Lomax distribution with four parameters to the survival time of breast cancer patients in Gaza Strip data and failure times of aircraft windshield. This chapter is divided into three major sections:  The models used for comparison with the proposed distributions.  The Data and the models' fits.  The truncated Data and the models' fits.

In Chapter 7 we present our conclusions, recommendations, and suggest some prospective topics for further research.

Chapter 2

Development of Lifetime Distributions

2.1. Introduction

Modeling and analysis of lifetime data is an important aspect of statistical work in a wide variety of scientific and technological fields. Several distributions have been proposed in the literature to model lifetime data by combining some useful lifetime distributions. Adamidis and Loukas (1998) introduced a two- parameter exponential-geometric (EG) distribution by compounding an exponential distribution with a geometric distribution. In the same way, the exponential Poisson (EP) and exponential logarithmic (EL) distributions were introduced and studied by Kus (2007) and Tahmasbi and Rezaei (2008), respectively.

The two-parameter Weibull and Gamma distributions are the most popular distributions used for analyzing lifetime data. The gamma distribution has wide applications other than that in survival analysis. However, its major drawback is that its survival function cannot be obtained in a closed form unless the shape parameter is an integer. This makes the Gamma distribution a little less popular than the Weibull distribution, whose survival function and failure rate have very simple and easy-to-study forms. In recent years the Weibull distribution has become rather popular in analyzing lifetime data because in the presence of censoring it is very easy to handle (Pal et al., 2006).

New extensions of the Weibull distribution were proposed to model bathtub-shaped failure rate since the Weibull distribution is not adequate for modeling phenomenon with non-monotone failure rates. Among these, we cite the exponentiated Weibull (EW) by Mudholkar et al. (1995), the additive Weibull by Xie and Lai (1995), the modified Weibull extension by Xie et al. (2002) and the modified Weibull (MW) distributions by Lai et al. (2003).

More recently, Bourguignon et al. (2014) introduced a family of univariate distributions with two additional parameters namely Weibull - G Family influenced by the Zografos- Balakrishnan-G class, which naturally extends the EG and EP distributions, respectively by Zografos and Balakrishnan (2009).

The distribution of failures over the lifetime of the product population is critically important to the MEMS reliability physicist (small, lightweight and power saving systems compared to conventional systems). Using these concepts, distribution functions can be developed and used for predictive purposes. A hazard rate that changes over the lifetime of the product, starting high, reducing, and increasing towards the end of the product life, is also termed the “bathtub curve” (see Fig. 2.1). The population will have items that fail within the first few weeks to months of the product lifetime (infant mortality) is termed the bathtub curve because of the shape of the curve itself. An ideal failure behavior is to eliminate the failures due to defects in the infant mortality portion of the curve through burn- in and/or defect reduction programs, and to not operate the product into the wear- out phase. The operational life is within the typically constant hazard rate section of the curve.

Fig. 2.1. The bathtub curve: the three stages over the device lifetime: high initial failure, constant failure rate over the useful lifetime, and increased failure rate as the devices age.

There are various time to failure distributions to statistically express population lifetime behavior. Three popular statistical reliability distributions are the Exponential, the Weibull and the Lognormal (Hartzell et al., 2011) but we will only concentrate here on the Weibull distribution.

This chapter is outlined as follows: We present a historical look at the Weibull and Lomax distributions in Section 2 and Section 3, respectively. In Section 4, we discuss a mixture representation of density functions of the Weibull distribution. Finally, we have a special look at the Weibull-Pareto distribution in Section 5.

2.2. The Weibull distribution

The Weibull distribution is a very popular distribution named by a Swedish physicist Waladdi Weibull. He applied this distribution in 1939 to analyze the breaking strength of materials. Since then, it has been widely used for analyzing lifetime data in reliability engineering. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter. The Weibull distribution is a widely used statistical models for studying fatigue and endurance life in engineering devices and materials. Many examples can be found among the aerospace, electronics, materials, and automotive industries. Recent advances in Weibull theory have also created numerous specialized Weibull applications. Modern computing technology has made many of these techniques accessible across the engineering spectrum (Aryal & Tsokos, 2011).

The pdf and cdf of the two-parameter Weibull model, as given by Hartzell et al. (2011), has the following forms:

( ) ( )

( )

where is the scale parameter, and is the shape parameter.

The survival function S(x) and the hazard rate function h(x) at time x for the Weibull distribution, according to Hartzell et al. (2011), are given by

( )

The pdf of the two parameter Weibull model can be described as in Fig. 2.2.

Fig. 2.2. The pdf of Weibull function in units of α, varying β.

The cdf of the two parameter of the Weibull model is given in Fig. 2.3 while Hazard function is in Fig. 2.4

Fig. 2.3. The cdf of Weibull Function in units of α, varying β.

Fig. 2.4. Hazard rate for the Weibull function.

The Weibull distribution model has some advantages and has been applied in many situations including the following ones: Gumbel (1958) showed that the Weibull distribution is identical to the type III (the smallest) Extreme-value distribution. This evidence indicates that the Weibull distribution is applicable to the smallest random time of failure from many independent competing time (Tobias & Trindade, 1986). In other words, if the item consists of many parts, and each part has the same failure time distribution, and the item fails when the weakest part fails, then the Weibull distribution is an acceptable model of that failure mode (Nelson, 1982). This is the so called "chain model", i.e. the strength of the chain is equal to the strength of the weakest link. Therefore, the Weibull distribution is said to be suitable for modeling "the weakest link in the chain", that is, the first item to fail among a group of items all of which have many possibilities of failing (Abernethy, 1994). Kao (1957) showed that the failures of electronic tubes meet the smallest weak chain model and, hence, follow the Weibull distribution.

The primary advantage of Weibull analysis is the ability to provide reasonably accurate failure analysis and failure forecasts with extremely small samples. Solutions are possible at the earliest indications of a problem without having to "crash a few more." Small samples also allow cost effective component

16 testing. For example, the Weibull tests are completed when the first failure occurs in each group of components, (say, groups of four bearings). If all the bearings are tested to failure, the cost and time required is much greater.

Another advantage of Weibull analysis is that it provides a simple and useful graphical plot of the failure data. The data plot is extremely important to the engineer and to the manager. The Weibull data plot is particularly informative as Weibull pointed out in Weibull (1951). Fig. 2.5 is a typical Weibull plot. The horizontal scale is a measure of life or aging. Start/stop cycles, mileage, operating time, landings or mission cycles are examples of aging parameters. The vertical scale is the cumulative percentage failed. The two defining parameters of the Weibull line are the slope, beta, and the characteristic life, eta. The slope of the line, β , is particularly significant and may provide a clue to the physics of the failure (Abernethy, 2006).

Figure 2.5. The Weibull Data Plot: Time-to-Failure.

The slope of the Weibull plot, beta ( β ), determines which member of the family of Weibull failure distributions best fits or describes the data. The slope β , also indicates which class of failures is present: β < 1.0 indicates infant mortality β = 1.0 means random failures (independent of age) β > 1.0 indicates wear out failures

The Weibull plot shows the onset of the failure. For example, it may be of interest to determine the time at which 1% of the population will have failed. Weibull called this the "B1" life. For more serious or catastrophic failures, a lower risk may be required, B0.1 (age at which 0.1% of the population fail) or even B0.01 life (0.01% of the population). Six-sigma quality program goals often equate to 3.4 parts per million allowable failure proportion. That would be B0.00034! These values are read from the Weibull plot. For example, on Figure 2.5, the B1 life is approximately 160 and the B5 life is 300.

2.3. The Lomax distribution

The Lomax distribution, also called “Pareto type II” (the shifted Pareto) distribution by Lomax (1954), has been quite widely applied in a variety of contexts. Although introduced originally for modeling business failure data, the Lomax model belongs to the family of decreasing failure rate as indicated by Chahkandi and Ganjali (2009). It has been used for reliability modeling and life testing (e.g., Hassan and Al-Ghamdi, (2009)), and applied to income and wealth distribution data, firm size (Harris, 1968; Atkinson and Harrison, 1978 and Corbelini et al., 2007). It has also found applications in the biological sciences and even for modeling the distribution of the sizes of computer files on servers (Holland et al., 2006). Some authors, such as Bryson (1974), suggested the use of this distribution as an alternative to the exponential distribution when the data are heavy-tailed.

The Lomax distribution can be motivated in a number of ways. For example, Balkema and de Haan (1974) showed that it arises as the limit distribution of residual lifetime at great. Dubey (1970) showed that it can be derived as a special case of a particular compound gamma distribution. Moreover, Tadikamalla (1980) related the Lomax distribution to the Burr family of distributions. On the other hand, the Lomax distribution has been used as the basis for several generalizations. For example, Ghitany et al. (2007) extended it by introducing an additional parameter using the Marshal and Olkin (1997) approach. Al-Awadhi and Ghitany (2001) used the Lomax distribution as a mixing distribution for the Poisson parameter and derived a discrete Poisson-Lomax distribution. Punathumparambath (2011) introduced the double-Lomax distribution and applied it to IQ data. The record statistics of the Lomax distribution have been studied by Ahsanullah (1991) and by Balakrishnan and Ahsanullah (1993). The implications of various forms of right-truncation and right-censoring are discussed by Myhre and Saunders (1982), Childs et al. (2001), Cramer and Schmiedt (2011) and others; and sample size estimation has been discussed by Abd-Elfattah et al. (2006).

The two-parameter Lomax distribution has the following pdf and cdf :

( ) ( )

where is the shape parameter, and is the scale parameter. In some applications it is useful to incorporate a location parameter, but we do not pursue that here.

The survival function S(x) and the hazard rate function h(x) at time x for the Lomax distribution are given by Lingappaiah (1987) as follows:

( ) ( )

respectively.

2.4. Mixture Representation of the Density Function of the Weibull Distribution

[ ] where is a non-negative monotonically increasing function depending on the parameter vector . The corresponding probability density function pdf becomes:

[ ] where is the derivative of . Different functions in (2.7) including some important statistical models such as : gives the exponential distribution, leads to the Rayleigh distribution; yields the Pareto distribution, and [ ] gives the Gompertz distribution.

Recently, Zografos & Balakrishnan (2009) proposed and studied a broad family of univariate distributions through a particular case of Stacy's generalized gamma distribution. Consider a continuous distribution with density , and further Stacy's generalized gamma density in the form for and positive parameters and . Based on this density, and by replacing x by [ ] and considering , Zografos & Balakrishnan defined their family with the following cdf

[ ]

where ∫ denotes the incomplete Gamma function and is the gamma function. This pdf family is given by

{ [ ]}

Bourguignon et al.(2014) introduced and studied in generality a family of univariate distributions with two additional parameters namely Family influenced by the Zografos- Balakrishnan-G class, in the same vein as the extended Weibull Gurvich et al. (1997) and gamma Zografos & Balakrishnan, (2009) families, using the Weibull generator applied to the odds ratio [ ]. The term "generator" means that for each baseline distribution we have a different distribution .

Consider a continuous distribution with a density and the Weibull cdf

with positive parameters . Based on this density, by replacing with [ ], we define the cdf family by

∫

{ * + }

where is a baseline cdf, which depends on a parameter vector . The family pdf reduces to

{ * + }

Henceforth, let be a continuous baseline distribution. For each distribution, we define the distribution with two extra parameters and defined by the pdf (2.12). A random variable X with pdf (2.12) is denoted by . The additional parameters induced by the Weibull generator are sought as a manner to furnish a more flexible

21 distribution. If β = 1, it corresponds to the exponential-generator. An interpretation of the Wei-G family of distributions can be given following Cooray (2006) in a similar context. Let Y be a lifetime random variable having a certain continuous distribution. The odds ratio that an individual (or component) following the lifetime Y will die (failure) at time X is . Consider that the variability of this odds of death is represented by the random variable X and assume that it follows the Weibull model with scale and shape . We can write:

( ) where is given by (2.11).

Alzaatreh (2011 and 2013b) and Alzaatreh et al. (2012) used the Transformed-Transformer (T-X) family of distributions to define the Gamma- Pareto (GaP) cdf by:

∫ where ( ) [ ] and ⁄ ⁄[ ]

It is noteworthy to mention that Eq. (2.11) is a special case of the T-X family proposed by Alzaatreh et al. (2013b) given in Eq. (2.14) by taking ⁄ ̅ and as the Weibull cdf.

2.5. Weibull-Pareto distribution

Based on the T-X family Alzaatreh (2011), Alzaatreh et al. (2013a) defined the Weibull-Pareto-T-X (WPTX) distribution. If a random variable Z has the Weibull distribution with parameters and , then the WPTX cumulative function is given by

* ( )+

Also, Tahir et al. (2010) proposed an extension of the Pareto model called the Weibull-Pareto (“WP” for short) distribution based on equation

[ ] 0 1 ̅ ̅

The proposed distribution is more flexible than the WPTX model (Alzaatreh et al., 2013a). For example, the hazard function shapes of the WP distribution can be constant, increasing, decreasing, bathtub and upside down bathtub. Alzaatreh et al. (2013a) noted a significant problem in estimating the parameters of the their distribution using maximum likelihood method. Further, they showed that the maximum likelihood estimates (MLEs) have considerable bias values. On the other hand, the MLEs of the WP parameters have a lower bias values (Tahir et al., 2010).

In the next chapter, we will define a special distribution of our interest, called a Weibull-Lomax distribution, aiming to overcome problems faced in the Weibull distribution. We will discuss some mathematical and structural properties of the proposed distribution and discuss the shape of its density and hazard functions. We will also conclude relations between this our distribution and other distributions.

Chapter 3

The four-parameter Weibull-Lomax distribution

3.1. Introduction

The Weibull distribution is a popular life time distribution model in reliability engineering. However, this distribution does not have a bathtub or upside–down bathtub shaped hazard rate function, which is why it cannot be utilized to model the lifetime of certain systems. Many authors introduced flexible distributions for modeling complex data and obtaining a better fit than the Weibull distribution. In an attempt to overcome to this shortcoming, we give a generalization of the classical Weibull distribution.

The main aim of this chapter is to define and study in details the mathematical properties of the new distribution and discuss the shapes of the density and hazard rate functions. We provide a mixture representation for its density function. The new distribution provides an extension of the Lomax distribution and solve the problems of flexibility of the Weibull distribution by adding more flexibility for the Weibull distribution using the Weibull-G family defined by Bourguignon et al. (2014). Therefore, we look at the combination of both Weibull and Lomax distributions in one compound distribution called the Weibull-Lomax (“WL” for short) distribution by adding two extra shape parameters to the Lomax model.

This chapter is outlined as follows: The new generalization to each of Weibull and Lomax distributions called WL distribution is introduced in Section 2. In Section 3, the transformations on the WL distribution including some relations of the WL distribution with the well-known Weibull and exponential distributions. The WL density function can be wrought as a double mixture of Exponentiated Lomax densities discussed in Section 4. Various properties including hazard rate function, the quantile function, moments, coefficient of variation, incomplete moments, Bonferroni and Lorenz curves, the Mean deviations, mean residual life,

Entropies and generating function are investigated in Section 5. Shapes of the limit of the density function and the hazard rate function, and shapes of the hazard rate function for some various and fixed value of parameters are discussed in Section 6.

3.2. The Weibull-Lomax distribution

To add more flexibility to the Weibull distribution many researchers developed many generalizations of this distribution. One such attempts was due to Bourguignon et al.(2014) who proposed the Weibull – G class influenced by the Zografas-Balakrishnon-G class. Let be the cumulative distribution function cdf of any random variable X with parameter vector Ԑ and be the probability density function pdf of a random variable T defined on [ . By replacing X with the odds ratio ̅ , (s.t: ̅ ) we define the cdf of the generalized family of distribution defined by Bourguignon et al. (2014) as :

̅ ∫

If a random variable T follows the Weibull distribution with parameters

, the definition in (3.1) leads to the Weibull- G family with the cdf :

∫

and

̅ ∫ ̅

( ) ̅ ̅̅̅ ̅ ̅̅ , which yields

( ) ̅

Then, its pdf is given by

̅ ( ) ( ( ) * ̅ ̅ ̅

( ) ( ) ̅ ̅ ̅

( ) ( ) ̅ . ̅ ̅

This yields:

( ) ̅ ̅

3.2.1. The Cumulative distribution function of the distribution

If X is a Lomax random variable with the pdf and cdf functions:

( ) ( )

From (3.2), we obtain the cdf of WL as

( ) 4 5

( * ( ) ̅

(( ) *

and we can reduces (3.5) to :

.( * /

where a > 0 and b > 0 are two additional shape parameters for Lomax distribution. Plots of the WL cdf for some parameter values are displayed in Fig 3.1.(a),(b).

Fig. 3.1. Plots of the WL cdf for some parameter values.

3.2.2. The probability density function of the

If X is a Lomax random variable with the pdf and cdf functions:

( ) ( )

From (3.3), we obtain the pdf of WL as

( ) ̅ ̅

( ) 4 5 ( ( ) * ( )

( )

( ( ) *

(( ) * ( ) ( ) ( ( ) )

(( ) * ( ) ( ( ) )

This yields:

(( ) * ( ) ( ( ) )

and we can reduces (3.7) as :

(( ) * ( ) (( ) )

where are two additional parameters. A random variable X with a cdf as in (3.6) and a pdf as in (3.8) is said to follow the four-parameter Weibull- Lomax distribution, and will be denoted by Plots of the WL pdf for some parameter values are displayed in Fig 3.2(a),(b).

Fig. 3.2. Plots of the WL pdf for some parameter values.

3.3. Transformations on the Weibull-Lomax distribution

Now, we can study the structural properties of this distribution because it extends some important distributions previously considered in the literature. Lemma 1 provides some relations of the WL distribution with the well-known Weibull and exponential distributions.

Lemma 1 (Transformation):

(a) If a random variable Y follows the Weibull distribution with shape parameter b and scale parameter one, then the random variable [ ⁄ ] follows the distribution.

(b) If a random variable Y follows the exponential distribution with mean , then

⁄ the random variable 0( ⁄ ) 1 follows the distribution.

Proof:

(a) The results follow by using the transformation of random variables techniques. From theorem 2.1.5 of Casella & Berger (2001) we have:

( ) | |

Let be the Weibull-Lomax pdf as in (3.8),

(( ) * ( ) (( ) )

where .

Let ( ) , then ( ⁄ )

and | | . By substituting the last two equations at (3.9)

with scale parameter , we get,

⁄ ( ( ))

From by replacing the variable by ( ⁄ ) and substituting it in the previous equation, we have

⁄ ⁄ ( ) ( ) . / .. / /

( ⁄ * 44 5 5

Thus,

( )

Then,

Therefore, if a random variable Y follows the Weibull distribution with shape parameter b and scale parameter one, then the random variable

[ ⁄ ] follows the distribution.

⁄ (b) Let (( ) ) , then .( ⁄ ) /

⁄ and | | ( ) . By substituting the last two

equations at (3.9) we get:

⁄

( .( ⁄ ) /+ ( ⁄ )

⁄ From by replacing the variable by .( ⁄ ) /and

substituting it in the previous equation, we have:

⁄ ⁄ (( ⁄ ) + (( ⁄ ) +

( , (( , ,

⁄ ⁄ 4. / 5

⁄ (( ) ) ( )

⁄ = ( )

then

Therefore, if a random variable Y follows the exponential distribution, then the

⁄ random variable 0( ⁄ ) 1 follows the distribution.

3.4. Annotation of the Weibull-Lomax distribution: Mixture representation of the WL pdf

Theorem 3.1.

The WL density can be wrought as a mixture representation of the Exponentiated Lomax density.

Proof:

The WL density function can be wrought as a double mixture of Exponentiated Lomax (EL for short) densities. In order to a simplify the form of the WL pdf, we expand (3.8) in a power series as,

( ) 4 5 ( ( ) * ( ( ) *

( )

. ( ( ) */

( ( ) *

( ) ( ) ,

. ( ( ) */

( * ( ,

. ( * / where

Then, the WL pdf can be expressed with A inserted in the last equation. After a power series expansion, the quantity A reduces to:

[ ( ) ]

∑ { }

0 [ ( ) ]1

Since the power series ∑ , combining the last result, we have:

( ( ) * [ ( ) ]

( ) ( ) ∑ { }

. ( ( ) */ 0 [ ( ) ]1

( ) ( ) ∑ ( ( ) )

where . ( ( ) )/

After a power series expansion, the quantity B reduces as follows:

Let , , ( )

∑ Since ( ) ,

then ∑ ( ) ( ( ) ) . ( ( ) )/ ,

∑ ( ) ( ( ) ) . ( ( ) )/ ,

∑ ( ( ) ) . ( ( ) )/ , ( )

Thus,

∑ ( ( ) ) . ( ( ) )/

Therefore,

∑ ( ( ) )

Combining the last result, we can rewrite,

( ) ( ) ∑ ( ( ) )

∑ ( ( ) )

∑

( ) ( ) ( ( ) ) ,

∑

( ) ( ( ) )

The last equation reduces in a more simplified form to:

∑ where:

( ) ( ( ) )

and .

We can observe that, ( ) ( ( ) ) which is the original form of the EL density (Abdul-Moniem & Abdel-Hameed (2012)), s.t in Equation (3.12) equals .

Equation (3.12) reveals that the WL pdf can be wrought as a double mixture of EL densities. So, many of its mathematical properties can be derived from the form of the EL distribution. Equation (3.12) is the main result of this section.

3.5. Mathematical structural properties of Weibull Lomax distribution

3.5.1. The Hazard rate function and other related functions of the Weibull Lomax ( ) distribution

Henceforth, a random variable X with density function (3.6) is denoted by , and its survival function hazard rate function , reversed-hazard rate function and cumulative hazard rate function of X can be established as follows:

3.5.1.1. The Survival function of the Weibull Lomax ( ) distribution

The associated with WL distribution is,

Substituting Eq. (3.6) at the previous equation we get:

(( ) * . /

(( ) *

.( * /

Survival data are data that describe the time to a particular event. This event is usually referred to as the failure of some machine or death of a person. However, survival data can also represent the time until a cancer patient relapses or time until another infection occurs in burn patients. The survival function of a positive random variable X defines the probability of survival beyond time x.

3.5.1.2. The Hazard rate function of the Weibull Lomax ( ) distribution

The associated with WL distribution is,

Substituting Eq. (3.8) and (3.13) in the previous equation we get:

.( * / ( )( ) (( ) *

.( * /

( ) ( ) (( ) ) ,

Thus,

( ) (( ) )

Many shapes of hazard rate functions has be shown below in Fig.3.3.

3.5.1.3. The reversed-hazard rate function (rhrf) function of the Weibull Lomax ( ) distribution

The associated with WL distribution is,

Substituting Eq. (3.8) and (3.13) in the previous equation yields :

.( * / ( ) (( ) *

(( ) *

Fig. 3.3. Plots of the hrf of WL for some parameter values.

3.5.1.4. The cumulative hazard rate function function of the Weibull Lomax ( ) distribution

The , of the WL distribution is,

*( ) +

Plots of the hrf of WL for some parameter values are displayed in Fig. 3.3.

Notes:

We can observe that, the WL distribution have sufficient flexibility for generating data. For example, when you look at Fig. 3.3.

 In Fig.3.3(a), for fixed values of and different values of , we can see a bathtub curve of hazard rate function.  In Fig.3.3(b), for fixed values of and different values of , we can see an upside down bathtub curve of hazard rate function.  In Fig.3.3(c), for fixed values of and different values of , we can see an increasing curve of hazard rate function.  In Fig.3.3(d), for fixed values of and different value of , we can see a decreasing curve of hazard rate function.  In Fig.3.3(e), for fixed values of and different values of , we can see a constant curve of hazard rate function.

3.5.2. Quantile distribution function

While the cumulative distribution function provides a complete specification of the properties of a random variable, it is useful to use simpler and more easily understood measures of the central tendency and range of values that a random variable may assume. Perhaps the simplest approach to describe the distribution of a random variable is to report the value of several quantiles. The

38 quantile of a random variable X is the smallest value such that X has a probability p of assuming a value equal to or less than .

[ ] [ ]

Equation is written to insist if at some value , the cumulative probability function jumps from less than p to more than p, then that value will be defined as the quantile even though . If X is a continuous random variable, then in the region where , the quantiles are uniquely defined and are obtained by the solution of ( ) (Loucks, et al., 2005). In terms of the distribution function F, the quantile distribution function Q returns the value x such that:

( ) ( )

Using this function, simulating a WL random variable becomes a straightforward procedure. Let p be a uniformly distributed variable on the unit interval ]. Using the inverse transformation method, by inverting (3.6), the quantile distribution function (qf) of X follows as

(( ) *

( )

.( * /

( ) ⁄

⁄ * ⁄ +

Let , then

⁄ ⁄ * +

Thus, a random variable satisfying the equation in (3.18) has a WL pdf given by (3.8), i.e., .

Special quantiles:

The quantile of order ⁄ is called a median (m) of the distribution. The median is frequently used as a measure of the center of the distribution. As a special case of (3.18) when p=0.5, is the median and can be simply defined as:

⁄ ⁄ * + .

The first quartile is defined as follows,

⁄ ⁄ * +

The quantile of order ⁄ is called the first quartile. The third quartile, however, is the quantile of order ⁄ and can be defined as follows,

⁄ ⁄ * +

The median is the second quartile. Assuming uniqueness, let

denote the first, second, and third quartiles of X, respectively. Note that the interval from to , is known as the interquartile range and gives the middle half of the distribution. Thus the interquartile range is defined by

3.5.3. Moments of the Weibull-Lomax distribution

The rth complete moment of the WL distribution X follows from (3.12) as:

To obtain of low order of r, such as r=1, r=2,... it is much easier to first obtain as follows:

∫

(See, El-Bassiouny, et al., 2015, p26 for the use of this function for the Exponential Lomax Distribution.) then,

∑ ∫

∑ ∫ ( ) ( ) (

( ) )

∑ ∫

. ( * /

∑ ∫ ( ) ( )

( ( ) )

∑ ∫

(( ) ) ( ) ( ( ) )

∑ ∫

(( ) ) ( ( ) )

Now, let ( ) then , and .

Applying the above transformation, the integral becomes,

( ) ∑ ∫

∑

( ) ∫

∑

( )

where ∫ is the complete beta function.

Then, the first four non-central moments of the WL distribution may be obtained from (3.19) as follow:

3.5.3.1. The Mean of the WL distribution

Setting r = 1 in (3.19), the mean of X reduces to

∑

( )

∑

( )

3.5.3.2. The second moment of the WL distribution

Setting r = 2 in (3.19), the second non-central moment of X reduces to:

∑ ( )

*∑ ( ) +

∑ ( *

∑ { ( *

( *}

3.5.3.3. The third moment of the WL distribution

Setting r = 3 in (3.19), the third non-central moment of X reduces to

∑ ( )

( ) ( )

∑ ( *

2 ∑ ( *

∑ ( * 3

2∑ ( * 3

∑ ( *

3.5.3.4. The Forth moment of the WL distribution

Finally, setting r = 4 in (3.19), the fourth non-central moment of X reduces to:

∑ ( *

∑

( *

∑ ( *

0 ∑ ( *

∑ ( *

∑ ( * 1

0 ∑ ( *

∑ ( * 1

0∑ ( * 1

∑ ( *

The variance, skewness and kurtosis measures can be calculated from the ordinary moments using well-known relationships.

3.5.3.5. The Variance of the Weibull Lomax distribution

The variance can be calculated from the ordinary moments using the well- known relationship:

∑ ( *

∑ ( )

*∑ ( ) +

∑ ( )

3.5.3.6. The Skewness of the Weibull-Lomax distribution

The Skewness measure can be calculated from the ordinary moments using well-known relationship

, ∑ ( ) ⁄ ( )

∑ ( )

∑ ( ) - ⁄

, ∑ ( )

⁄ ∑ ( ) -

3.5.3.7. The Kurtosis of the Weibull Lomax distribution

The Kurtosis measure can be calculated from the ordinary moments using well-known relationship

=, *∑ ( )

∑ ( *

∑ ( * 1 ⁄

0 ∑ ( *

∑ ( ) + } - 3

3.5.4. Coefficient of Variation

The Coefficient of Variation (CV) is expressed as the ratio of standard deviation and mean. The CV is the measure of variability of the data. When the value of the CV is high, it means that the data has high variability and less stability. When the value of the CV is low, it means that the data has less variability and high stability. The formula for the CV of WL distribution is given in below:

*∑ ( )

⁄

∑ ( ) ] ⁄

*∑ ( ) +

3.5.5. Incomplete moments

Theorem 3.2.

The rth incomplete moment of a random variable X having the WL distribution, say , is given by substituting r=1, r=2,...... , in the following equation,

∑

* (( ) ) +

Proof:

To obtain the rth incomplete moment of X , we first evaluate the following moment defined by

∫

∑ ∫

∑ ∫ ( ) ( ) (

( ) )

∑ ∫ . ( * /

∑ ∫ ( ) ( )

( ( ) )

∑ ∫

(( ) ) ( ) ( ( ) )

∑ ∫

(( ) ) ( ( ) )

Let ( ) then ,

( ) .

( ) ∑ ∫

( )

( ) ( ) ∑ ∫

∑

( ) ( ) ( ) 0 1 ∫ ∫

By using the incomplete and complete beta functions :

∫

, ∫ Then (3.28) implies that

∑

* (( ) ) +

∑

* (( ) )+

The first incomplete moments of the Weibull-Lomax distribution where obtain from of low order of r, such as r=1, r=2,...... , it is much easier to obtain it by substitute it in .

For Ex., at r=1, we have ∑

* (( ) )+

then, ∑

* (( ) )+

3.5.6. Bonferroni and Lorenz Curves

Bonferroni and Lorenz curves have many applications not only in economics to study income and poverty, but also in other fields like reliability, demography, insurance and medicine. They are defined by

∫ and

∫

respectively, where and .

From ∫ , or ∫ we have

and (Paranaiba, et al., 2010).

3.5.7. The moment generating function

Let be a random variable following the The of X, say [ ] may be obtained by first computing the of as follows,

( )

∫

∫ ∑

∑

where follows from equation .

∑ ∑ ( *

∑ ( *

Equation gives the of for the distribution.

Now, since we are interested in obtaining , the of X for the

distribution and is defined by ( )

, we can provide a simple representation for the of the distribution by setting for ,

( ) ,

∑ ( *

3.5.8. Mean deviations

The mean deviations about the mean and the median can be used as measures of spread or scatter in a population from the center. Let and

be the mean and the median of the WL distribution, respectively. Ashour & Eltehiwy (2013) calculated the mean deviations about the mean and about the median as:

∫ and

∫

respectively. We may obtain

∫

and ∫ where denotes the first incomplete moment calculated from for r=1, then,

∑

[ (( ) *]

which yields:

∑ * ( )

..∑ ( * / /1

Moreover, from (3.6) the cdf of for WL distribution we have,

.( * /

which yields

..∑ ( )/ /

Substituting equations we have the mean deviations about the mean . Similarly, Substituting equations

we have the mean deviations about the median .

3.5.9. Mean residual life

The Mean Residual Life (MRL) function is of particular interest because of its easiness of interpretability and large area of applications (Guess & Proschan, 1985). The mrl function computes the expected remaining survival time of a subject given survival up to time X (Poynor, 2010). Suppose that and

∫ then the mrl function for continuous X is defined as:

∫ ∫ ⟨ ⟩

Now, assuming that with survival function as in , the MRL function is defined as the expected value of the remaining lifetimes after a fixed time point x, i.e.

.( * /

∫

.( * /

This integral is indefinite but the mean residual life could be computed numerically.

3.5.10. Entropies

An entropy is a measure of variation or uncertainty of a random variable X. Two popular entropy measures are the Rényi and Shannon entropies (Rényi, 1961 and Shannon, 1948). Here, we define the Rényi entropy and the Shannon entropy of a random variable X of the WL distribution. The Rényi entropy of a random variable with pdf is defined as:

.∫ /

for and . The Shannon entropy of a random variable X is defined by [ ]. Shannon showed important applications of this entropy in communication theory. Many other applications have also been used in different areas such as engineering, physics, biology and economics. It is the special case of the Rényi entropy when . These are easily obtained as follows. (for details, see for example, Nadarajah, 2005).

(i) Rényi Entropy: Following Rényi (1961), for some real values and , the entropy of the WL random variable X having the pdf is given by

.∫ /

(( ) * (∫ [ ( ) (( ) ) ] +

This integral is so difficult to obtain, but it can be computed numerically for most parameter values.

(ii) Shannon Entropy : Following Shannon (1948), the entropy of the WL random variable X having the pdf is given by

[ ] [ ( )] ∫ ( )

which is also the particular case of Rényi entropy as obtained in (i) above for .

3.5.11. The Mode of the WL ( ) distribution

The mode of the WL ( ) distribution is the value x at which its pdf has its maximum value. That is,

(( ) * [ ( ) (( ) ) ]

.( * / 6 0( * 1 .( * / ( *

.( * / [.( * / ] ( * .( * /

.( * / 6 7]⁄

Thus,

(( ) * [ ( * ( * .( * /

.( * / ( * .( * / 0( * 1

.( * / [ .( * / ] ( * .( * / 7⁄

.( * / ( ) (( ) * 0 ( ) (( ) * ( ) 1

.( * / ( * .( * /

0 ( ) (( ) ) ( ) +

This means that either, ( ) (( ) )

(( ) *

[ ( ) (( ) ) ( ) ]

Since, then all roots of ( ) (( )

(( ) * ) are rejected, then,

[ ( ) (( ) ) ( ) ]

i.e. ( ) (( ) ) ( )

( ) [ (( ) ) ]

Therefore, we get the mode of the Weibull Lomax ( ) distribution numerically using (3.34).

3.6. Shapes of the density and hazard rate functions

3.6.1. The limits of the density function and the hazard rate function

Theorem 3.3.

The limit of the WL pdf as is 0. Also, the limits of the pdf and hrf of X as x are given by,

{

Further, the limits of the hrf of X as are given by

{

Proof: Most results of this type can be easily shown from equations (3.6) and (3.14). We only need to obtain the result in (3.36). To do this, one can see from the hrf in (3.14) that

( ) (( ) )

( ) (( ) ) ( ) ( )

( )

( ) . / ( )

( ) ( ( ) )

This implies that:

8( * . ( * /9

2( ) ( ( ) )3

2( ) ( ) 3

2 ( ) ( ) 3 2 ( ) 3

( *

This implies the result in (3.36).

3.6.2. Shapes of the hazard rate function

Theorem 3.4. The hrf of X possesses the following shapes: i. Constant failure rate (CFR) wherever ii. Increasing failure rate (IFR) wherever iii. Decreasing failure rate (DFR) wherever iv. Upside-down bathtub rate (UBT) wherever v. Bathtub rate (BT) wherever

Proof:

Based on Eq. (3.37), the hrf of X can be expressed as

2( ) ( ( ) )3

which implies the results (i) ,(ii) , (iii). Now , the derivative of in (3.14):

( ) (( ) )

is given by

[ ( ) (( ) ) ]

[( ) (( ) ) ]

[(( ) ) ( )

( * .( * / ]

[(( ) ) ( ) ( )

( * .( * / .( * /]

[(( ) ) ( )

( * .( * / ( * ]

( * ( * .( * /

( * .( * /

( ) [ ( ) (( ) )

( ) (( ) ) ]

( ) ( ) (( ) )

0 .( * / ( * 1

( ) ( ) (( ) )

0 .( * / ( * 1

( * .( * /

* ( ) +

( * .( * /

* ( ) +

( ) (( ) )

where ( ) .

Now,

( ) (( ) )

and , ( ) and (( ) ) doesn't equal zero,

Since , then, ( ) .

⁄ The critical value of h(x) is at ( ) , which is defined only when

Now, if then is defined and

and This proves (iv).

Similar arguments can be used to prove (v). If

This proves (v).

Note:

The text of a previous theorem compatible with our notes at the end of section 3.5.1.

3.7. Summary

In the this chapter we discussed some properties of the WL ( ) distribution and obtained many formulae such as:

 The probability density function and the Cumulative distribution function.  Transformations on the Weibull-Lomax distribution.  Mixture representation of the WL pdf .  The WL density function can be wrought as a double mixture of

Exponentiated Lomax densities, as: ∑ ,

where ( ) ( ( ) ) is the original form of

Exponentiated Lomax.  The Survival function of the Weibull Lomax( ).  The Hazard rate function (hrf) of the Weibull Lomax ( ).  The reversed-hazard rate function (rhrf) (r(t)) .

 The cumulative hazard rate function (chrf) (H(t)).  It is noted that there are different shapes of hazard rate function such as bathtub, upside down bathtub, increasing, decreasing and constant curve.  The quantile function (qf) of the Weibull Lomax ( ).  As a special cases, from the quantile function the median, the first quartile and the third quartile were obtained.  The rth moments of the Weibull-Lomax and the first four non-central

moments , variance, coefficient of variation, mean deviations, skewness and kurtosis of the Weibull-Lomax distribution were obtained.  The rth incomplete moments, Bonferroni and Lorenz Curves were obtained.  The moment generating function, the Mean Residual Life (MRL) function, the Rényi entropy and the Shannon Entropy were defined.  The mode of the Weibull Lomax( ) distribution was derived.  Shapes of the density function and the hazard rate function and the limits of the pdf & hrf when x close to infinity and zero from right side were obtained.

In the next chapter, in finding the MLE for the parameters of the Weibull- Lomax distribution, we are going to find the first derivative of the log-likelihood function, and accordingly we find the second derivative of log-likelihood function. Based on the above, we will give the observed Fisher information matrix, to derive the variance, and the standard error for the estimated parameters. We can then use this for building a confidence interval. Finally, we will discuss results of a simulation study and use it to estimate the parameters and their standard errors.

Chapter 4

Estimation of the parameters of the Weibull-Lomax

distribution

4.1. Introduction

The objectives of this chapter are to explore the maximum likelihood estimators (MLE) of the parameters of the Weibull-Lomax distribution. The idea of maximum likelihood estimation is perhaps the most important concept in parametric estimation. It is straightforward to describe (if not always to implement); it can be applied to any parametric family of distributions (that is, any class of distributions that is indexed by a finite set of parameters ).

In this chapter, we will derive the maximum likelihood estimates (MLEs) for the four parameters in section 2. We will then obtain the information matrix and the confidence intervals of the model parameters from a sample of observations in section 3. Finally, in section 4 we will present and discuss a simulation study on estimates of parameters of the underlying Weibull-Lomax distribution.

4.2. Maximum Likelihood Estimators of the parameters of the Weibull-Lomax distribution:

The maximum likelihood estimation (MLE) method has been a popular choice of model fitting in many fields (Rubin, et al., 1999; Lamberts, 2000; and Usher & McClelland, 2001) and it is a standard approach to parameter estimation and inference in statistics. MLE has many optimal properties in estimation: sufficiency, consistency, efficiency, and parameterization invariance. Further, many of the inference methods in statistics are developed based on MLE. For example, MLE is a prerequisite for the chi-square test, the G-square test, Bayesian methods, inference with missing data, modeling of random effects, and many

62 model selection criteria such as the Akaike information criterion (Akaike, 1973) and the Bayesian information criteria (Schwarz, 1978 and Myung, 2003).

Let be n observed values from the WL distribution with parameters Let be the parameter vector. If

is the log-likelihood based on a sample of n independent observations, with parameter vector , The log-likelihood function obtained from the density function of the WL distribution for the parameters vector is given by:

∏

First, we can easily define the likelihood function from as follows,

(( ) * ∏ [ ( ) (( ) ) ]

(( ) * ( ) ∏ ( ) ∏ (( ) ) ∏

Thus, we have,

∑ ( )

(( ) * ∑ (( ) ) ∑

∑ ( )

(( ) * ∑ (( ) ) ∑

∑ ( )

∑ (( ) ) ∑ (( ) )

The log-likelihood function can be maximized either computationally using the R library (AdequecyModel), SAS (PROC NLMIXED) or the Ox program (sub- routine MaxBFGS) (Doornik, 2007) or mathematically through solving the

63 nonlinear likelihood equations derived through differentiating with respect to the parameters (Hamedani, et.al., 2014).

The score function ( ⁄ ⁄ ⁄ ⁄ ) has components given by

∑ (( ) )

∑ (( ) ) (( ) )

( ) ( )

∑ ( ) ∑ ( )

∑ (( ) ) (( ) )

( ) ( )

∑ ( ) ∑ ( )

∑ (( ) ) ( *( ) + [ ])

( ) ( )

∑ ( ) ∑ ( )

∑ (( ) ) (( ) ( ) )

( ) ( )

∑ ( ) ∑ ( )

∑ (( ) ) ( ) ( )

After some algebra steps, the last formula reduces as follow,

[(( ) * ] ( )

∑ ( ) ∑ ( )

∑ (( ) ) *(( ) ) + ( )

∑ ( ) ∑ 0 1 ( ) ( )

∑ [(( ) ) (( ) ) ] ( )

∑ ∑ ( ) ( ) (( ) *

∑ (( ) ) ( )

Setting the above equations to zero and solving them simultaneously yield the MLEs of the four parameters. However, there is no closed-form solution to the likelihood equations and , and a suitable numerical algorithm must be adopted to obtain the MLEs of the parameters and

4.3. Confidence interval estimates of the parameters

To derive the confidence interval estimates for the model parameters vector

, we require the expected information matrix [ ] . Under certain regularity conditions, the maximum likelihood estimators asymptotically have (for large samples) a multivariate normal distribution (Lindgren, 1976, p. 272) with mean equals the true parameter value and variance-covariance matrix given by the inverse of the information matrix, i.e.

. This result can be used to construct approximate confidence intervals for the model parameters.

Since the information matrix depends on , Efron and Hinkley (1978) recommended that the observed Fisher information matrix ̂ may be used instead of the expected Fisher information matrix in the estimation of the variance of the maximum likelihood estimator, where the observed Fisher information matrix (evaluated at ̂) is given by:

( ̂) 0 1

The following equations are the elements of observed information matrix ( ̂)

∑ (( ) ) (( ) )

∑ ( ) ( ) (( ) )

( ) ( ) ∑

(( ) *

( ) ( ). (( ) * (( ) * (( ) * /

( )

( ) ( ) (( ) )

( ) ( )

(( ) *

( ) (( ) *

( ) ( )

(( ) * ( )

.(( ) *( ( ) ( ) ( ) * ( ) ( )/

(( ) *

. ( ) ( )(( ) * / ( ) (( ) *

( ) ( )(( ) *

( )

(( ) * ( ) ( )

( ) (( ) *

(( ) * ( ) ( )

( ) (( ) *

(( ) * ( ) ( )

(( ) *

Then, the confidence intervals for the parameters are given by:

̂ √ ̂ , ⁄

̂ √ ̂ , ⁄ ̂ √ ̂ and ⁄

̂ √ ̂ ⁄

respectively. Since ( ̂) is the observed information

[ ] matrix and we need its inverse " ̂ " to estimate the variance, then is given by the diagonal elements of ̂ corresponding to the four model parameters, which the multiplying of the main diagonal doesn't equal to zero, and

is the quantile ⁄ of the standard normal distribution. ⁄

4.4. Simulation Study

Simulation provides a powerful technique for answering a broad set of methodological and theoretical questions and provide a flexible framework to answer specific questions relevant to one’s own research. For example, simulation can evaluate the robustness of a statistical procedure under ideal and non-ideal conditions, and can identify strengths (e.g., accuracy of parameter estimates) and weaknesses (e.g., type-I and type-II error rates) of competing approaches for hypothesis testing and evaluating complicated standard errors of estimators.

Simulation can be used to estimate the statistical power of many models that cannot be estimated directly through power tables and other classical methods (e.g., mediation analyses, hierarchical linear models, structural equation models, etc.). The procedures used for simulation studies are also at the heart of bootstrapping methods, which use resampling procedures to obtain empirical estimates of sampling distributions, confidence intervals, and p-values when a parameter sampling distribution is non-normal or unknown (Hallgren, 2013).

Table 4.1. Estimated MLEs, SEs of the MLEs of parameters of WL distribution based on 1000 simulations with n=25,

50, 200, 500 and 1000.

Sample Actual values Estimated values Standard errors size ̂ ̂ ̂ ̂ ̃ ̃ ̃ ̃ 0.5 4.5 2.7 0.3 0.4018 4.4004 2.8020 0.3041 0.0002 0.0001 0.0004 0.0120 0.2 2.0 0.1 0.5 0.2135 1.8864 0.0914 0.5611 0.2831 0.0233 0.0137 0.0890 2.9 1.4 1.2 1.5 2.7955 1.4884 1.2902 1.6041 0.0134 0.1994 0.0500 0.0531 3.0 5.4 0.3 1.3 3.1007 5.5024 0.2844 1.1972 7.7e-5 3.3e-4 7.8e-3 4.1e-4 25 0.9 2.5 0.7 6.0 0.9643 2.4057 0.6505 5.8857 0.0156 0.0869 0.0507 0.0346 4.3 0.6 3.8 0.5 4.4000 0.7007 3.7000 0.5020 4.8e-5 1.4e-3 6.6e-5 1.6e-1

4.9 4.0 1.2 1.5 4.9974 4.3237 1.0212 1.2319 0.0092 0.5725 0.2713 0.3958 3.9 2.4 0.2 0.5 4.0005 2.5068 0.2016 0.5927 0.0002 0.0019 0.0156 0.0176

0.5 4.5 2.7 0.3 0.4018 4.4004 2.8019 0.3068 1.1e-4 4.4e-5 1.1e-4 8.0e-3 0.2 2.0 0.1 0.5 0.1779 1.8899 0.0937 0.5704 0.2214 0.0147 0.0109 0.0718 2.9 1.4 1.2 1.5 2.7969 1.4393 1.2923 1.6000 0.0119 0.1800 0.0415 0.0535 3.0 5.4 0.3 1.3 3.1007 5.5025 0.2852 1.1972 8.4e-5 4.9e-4 7.3e-3 6.5e-4 50 0.9 2.5 0.7 6.0 0.9636 2.3918 0.6389 5.8907 0.0060 0.0261 0.0276 0.0096 4.3 0.6 3.8 0.5 4.4000 0.7003 3.7000 0.4733 1.1e-5 6.2e-4 3.0e-5 1.0e-1 4.9 4.0 1.2 1.5 4.9985 4.2395 1.0547 1.2916 0.0085 0.4721 0.2222 0.3246 3.9 2.4 0.2 0.5 4.0006 2.5064 0.2087 0.5939 0.0001 0.0008 0.0186 0.0017

0.5 4.5 2.7 0.3 0.4017 4.4004 2.8019 0.3082 6.0e-5 2.0e-5 6.5e-5 3.8e-3 0.2 2.0 0.1 0.5 0.1509 1.8913 0.0944 0.5747 0.1608 0.0168 0.0092 0.0800 2.9 1.4 1.2 1.5 2.7987 1.4009 1.2975 1.5959 0.0018 0.0720 0.0055 0.0083 3.0 5.4 0.3 1.3 3.0981 5.5289 0.2825 1.1615 0.0056 0.0598 0.0095 0.0816 200 0.9 2.5 0.7 6.0 0.9512 2.4274 0.6381 5.8726 0.0164 0.0370 0.0187 0.0194 4.3 0.6 3.8 0.5 4.4000 0.7001 3.7000 0.4274 7.0e-7 1.6e-4 8.4e-6 5.2e-2 4.9 4.0 1.2 1.5 5.0004 4.1157 1.1097 1.3820 0.0011 0.0619 0.0301 0.0438 3.9 2.4 0.2 0.5 4.0006 2.5057 0.2075 0.5947 1.0e-5 1.9e-4 1.3e-3 5.9e-4

Table 4.1. Continued: Estimated MLEs, SEs of the MLEs of parameters of WL distribution based on 1000 simulations

with n=25, 50, 200, 500 and 1000. Sample Actual values Estimated values Standard errors size ̂ ̂ ̂ ̂ ̃ ̃ ̃ ̃ 0.5 4.5 2.7 0.3 0.4018 4.4004 2.8019 0.3061 6.3e-5 1.5e-5 6.9e-5 3.3e-3 0.2 2.0 0.1 0.5 0.1965 1.8898 0.0896 0.5637 0.2261 0.0121 0.0104 0.0753 2.9 1.4 1.2 1.5 2.7991 1.3870 1.2986 1.5935 0.0009 0.0468 0.0028 0.0061 3.0 5.4 0.3 1.3 3.1007 5.5024 0.2834 1.1972 1.6e-5 7.0e-5 1.6e-3 8.6e-5 500 0.9 2.5 0.7 6.0 0.9718 2.4046 0.6771 5.8881 0.0097 0.1025 0.0137 0.0451 4.3 0.6 3.8 0.5 4.4000 0.7000 3.7000 0.4077 1.6e-7 5.0e-5 2.5e-6 2.3e-2 4.9 4.0 1.2 1.5 5.0004 4.1146 1.1099 1.3832 0.0002 0.0056 0.0050 0.0074 3.9 2.4 0.2 0.5 4.0006 2.5065 0.1991 0.5935 2.6e-5 3.5e-4 2.9e-3 5.8e-4

0.5 4.5 2.7 0.3 0.4017 4.4004 2.8019 0.3082 2.6e-5 8.9e-6 2.6e-5 1.7e-3 0.2 2.0 0.1 0.5 0.1205 1.8938 0.0965 0.5866 0.0946 0.0108 0.0074 0.0518 2.9 1.4 1.2 1.5 2.7991 1.3912 1.2985 1.5929 0.0008 0.0458 0.0021 0.0053 3.0 5.4 0.3 1.3 3.0990 5.5192 0.2834 1.1747 0.0021 0.0225 0.0030 0.0305 1000 0.9 2.5 0.7 6.0 0.9429 2.4418 0.6312 5.8640 0.0166 0.0324 0.0209 0.0181 4.3 0.6 3.8 0.5 4.4000 0.7000 3.7000 0.4024 4.3e-8 2.7e-5 1.4e-6 1.2e-2 4.9 4.0 1.2 1.5 5.0004 4.1141 1.1099 1.3836 0.6773 0.0968 0.0858 0.0738 3.9 2.4 0.2 0.5 4.0006 2.5057 0.2073 0.5947 1.9e-6 3.6e-5 4.3e-4 2.4e-4

We evaluated the performance of the maximum likelihood estimates for the WL parameters using Monte Carlo simulation for a total of four parameter combinations. Different sample sizes (n = 25, 50, 200, 500 and 1000) are considered. The process is repeated 1000 times in order to obtain average estimates of the standard error (SE). They are listed in Table 4.1. The small values of the SEs indicate that the maximum likelihood method performs quite well in estimating the model parameters of the proposed distribution.

Looking carefully at Table 4.1., we note that the estimated values are very close to the actual values, and this is clear from the standard error of the estimated parameters, as they are very small. Also, whenever, the sample size increased further, the estimated values get closer to the actual values. In contrast, the standard error value decreases more and more. This means that the estimators are consistent.

4.5. Summary

In this chapter, we derived the maximum likelihood estimates (MLEs) for the four parameters of the WL distribution, but, we did not achieve a closed form for the estimate of each parameter. We showed that we can overcome this problem by solving the system of equations numerically. In the second part we obtained the elements of information matrix and then the confidence intervals of the model parameters from a sample of observations. Finally, we presented the results of a simulation study concerning the estimates of the parameters of the underlying distribution. We estimated the four parameters of the WL distribution, and we noted that when, the sample size increased, the estimates get closer to actual values. In contrast, the standard errors decrease further.

In the next chapter, we will define the Truncated Weibull-Lomax distribution, in order to have a new extension of the WL distribution. We will also make inference on our Truncated distribution, and present the mathematical and structural properties of the Truncated Weibull-Lomax distribution.

Chapter 5

Inference on the four-parameter truncated Weibull-Lomax distribution

5.1. Introduction

A truncated distribution is defined as a conditional distribution that results from a parent distribution and restricted to a smaller region. It occurs when there is no ability to know about or record events that occurs above or below a threshold set or outside a certain range. Truncation is different than censoring. With censoring, knowledge of items outside the restricted range is retained, but the full information could not be recorded. With truncation, knowledge of items outside the restricted range can't be obtained.

Let be a random variable from a distribution with a probability density function , a cumulative distribution function , and the range of the support . The density function of given the restriction that is defined by Hattaway (2010) as:

In this chapter, the study is limited to the doubly truncated four-parameter Weibull-Lomax distribution ( ), (“T-WL” for short), and inference is made through the above transformation. This chapter is outlined as follows. The T- WL distribution is introduced and discussed in Section 2. In Section 3, special cases of the T-WL distribution including the right truncated and the left truncated distributions are presented. Various mathematical properties including Hazard rate function, the quantile function, moments, coefficient of variation, the mode, incomplete moments, Bonferroni and Lorenz Curves, the mean deviations, mean residual life and generating function are investigated in Section 4.

5.2. The distribution function of the four-parameter T-WL distribution:

In this section, we define the pdf of the doubly T-WL distribution with four parameters ( ) and its properties including its cdf.

5.2.1. The probability density function of the doubly truncated

Let be a random variable having the four-parameter doubly truncated distribution and taking values in the interval [ ]. The truncated pdf of any variable may be written in the form:

∫

Then, by substituting (3.8) & (3.6) of the pdf and cdf respectively in (5.1), the four-parameter doubly truncated distribution function takes the form:

.( * / ( ) (( ) *

.( * / .( * /

.( * / ( ) (( ) *

.( * / .( * /

Therefore, the four-parameter truncated distribution and has the form:

(( ) * ( ) (( ) )

where the constant is such that:

.( * / .( * /

Or, we can rewrite as follows:

where is the pdf of the T-WL and is the pdf of the WL distribution.

Fig. (5.1) presents the shape of the doubly truncated distribution function using formula (5.3) with fixed left truncation point (c=0.6) and different values of right truncation point at (d=1.0, 1.2, 1.4, 1.6) together with the original distribution in the (black solid) line. Note that the greater the value of d the closer the curve of truncated distribution is from the curve of the original distribution.

Fig.5.1. Different density functions of fixed value of left truncation and various values of right truncation point

Figure (5.2) also presents the shape of the truncated distribution function using formula (5.3) with fixed right truncation point (d=1.5) and different values of left truncation point (c=0.6,0.8,1.0) together with the original distribution in a (black solid) line. Note that the greater the value of the further away the truncated distribution's curve is from the curve of the original distribution.

Fig.5.2. Different density functions of fixed value of right truncation and various values of lift truncation point 5.2.2. The Cumulative distribution function of the doubly truncated distribution

The cdf of the four-parameter doubly truncated distribution takes the form:

∫

(( ) * ∫ ( ) (( ) )

( ) ⁄ ⁄

and ⁄ ( ) .

When then ( ) and

.( * / ∫ ( * ( * .( * /

hence,

( *

∫ ⁄ ⁄

( *

∫

( *

Let .

When ( ) then ( ) and

as ( ) then ( )

Therefore,

( )

∫ ( )

Now, let ,

and we have that .

When ( ) then (( ) )

and as ( ) then (( ) ) .

Thus,

(( ) *

∫ .

(( ) *

Therefore,

Recall that

and that , then

Moreover, we have ( ) , then

(( ) * .

Thus,

.( * / ∫ ( * .( * /

(( ) * = | ,

.( * / .( * / 8 9

Therefore, the cdf of for ) is,

.( * / .( * / 8 9 where is a constant defined in the form (5.4), and are positive numbers. A random variable that have a cdf in Eq. (5.6) is called the truncated

Weibull Lomax (T-WL) distribution. Hereafter, we denote a random variable with cdf as in Eq. (5.6) by .

Figure (5.3) presents the shape of the cumulative distribution function of the truncated Weibull Lomax (0.5, 5, 4, 6) distribution using formula (5.6) compared with the cumulative distribution function of the original distribution. Note that the cumulative distribution function of the truncated Weibull Lomax ( ) is less extensive and prevalent than the cumulative distribution function of the original distribution.

Fig.5.3. Cumulative distribution functions of the truncated at different points

5.3. Special cases of the truncated Weibull Lomax ( )

Two special cases for the truncated Weibull Lomax distribution, may be obtained through the following:

5.3.1. A special case: The left truncated Weibull Lomax ( )

Let be a random variable having the truncated Weibull Lomax ( ) distribution and taking values in the interval [ . The pdf of any truncated variable having the WL distribution takes the form (5.3). Substituting in (5.4),

.( * / ( ) (( ) *

.( * / .( * /

.( * / ( ) (( ) *

.( * / .( * /

(( ) * then the amount tends to zero, and we get:

.( * / ( ) (( ) *

.( * /

i.e.

where, are positive numbers. A random variable having a pdf as in Eq. (5.7) is called the left truncated Weibull Lomax distribution. We denote a random variable with cdf by .

5.3.2. A special case: The right truncated Weibull Lomax ( )

Let be a random variable having the truncated Weibull Lomax ( ) distribution and taking values in the interval ]. The pdf of

78 this variable takes the form (5.3). Substituting in (5.4), then the amount

(( ) * tends to one, and we get:

.( * / ( ) (( ) *

.( * / .( * /

.( * / ( ) (( ) *

.( * /

i.e.

where, are positive numbers. A random variable having a pdf in the form (5.8) is called the right truncated Weibull Lomax distribution. We denote any random variable with cdf as in the form (5.8) by .

5.4. Mathematical properties of the truncated Weibull Lomax distribution

5.4.1. The Hazard rate function and other related functions of the truncated Weibull Lomax ( ) distribution

A random variable X with the density function in the form (5.3) has a survival function hazard rate function and reversed-hazard rate function of X .

5.4.1.1. The Survival function of the truncated Weibull Lomax ( ) distribution

Survival data describes the time to a particular event. This event is usually referred to as the failure of some machine or death of a person. Survival data can also represent the time until a cancer patient relapses or time until another infection occurs in burn patients. The survival function of a positive random variable X defines the probability of survival beyond time x. The survival function associated with distribution is,

Substituting the form in the last equation we get:

(( ) * (( ) * ( { }+

5.4.1.2. The Hazard rate function of the truncated Weibull Lomax ( ) distribution

The hazard rate function associated with the distribution is given by,

Substituting (5.3), (5.9) in the last equation we get :

.( * / ( ) (( ) *

.( * / .( * /

( { })

5.4.1.3. The reversed-hazard rate function (rhrf) function of the truncated Weibull Lomax ( ) distribution

The reversed-hazard rate function r(x) associated with the distribution is,

Substituting the forms (5.3), (5.9) in the previous equation we get:

.( * / ( ) (( ) *

.( * / .( * /

{ }

.( * / ( ) (( ) *

.( * / .( * /

5.4.2. The moments of the truncated Weibull Lomax distribution

The moment

The moment of the doubly truncated Weibull Lomax ( ) distribution follows the from (3.12) as:

To obtain of low order of r, such as r=1, r=2,...... , it is much easier to first obtain as follows:

∫

Since from (5.5), then

∫

Inserting the form (3.12) in Eq. (5.12), then Eq. (5.12) reduces to,

∑ ∫

∑ ∫ ( ) ( ) ( ( ) )

∑ ∫ ( ( ) )

∑ ∫ ( ) ( ) ( ( ) )

∑ ∫ (( ) ) ( ) ( ( ) )

∑ ∫ .

(( ) ) ( ( ) )

Let ( ) then , ( )

when and ( ) . Applying this transformation, the integral becomes,

∑ .

( ) ( )

∫ ( )

∑

( ) ( )

∫ ( )

∑

( ) ( ) 0 ∫

( ) ( ) 1 ∫

∑ 0 .( ) ( ) /

(( * ( ) +]

where is the incomplete beta function and given by,

∫

(See, El-Bassiouny, et al. (2015) for the use of this function for the Exponential Lomax Distribution.)

The first four non-central moments of the truncated Weibull-Lomax distribution follow below:

5.4.2.1 The Mean of the truncated Weibull Lomax distribution

Setting r = 1 in (5.13), the mean of X may be achieved as follows:

∑

0 .( ) ( ) / .( ) ( ) /1

∑

0 .( ) ( ) / .( ) ( ) /1

5.4.2.2. The second moment of the truncated Weibull Lomax distribution

Setting r = 2 in (5.13), the second non-central moment of X may be achieved as follows:

∑

[ (( * ( * +

(( * ( * +]

Hence,

∑

.0 .( ) ( ) /

(( * ( * +]

∑

.0 .( ) ( ) /

.( ) ( ) /1

2 ∑ 0 .( ) ( ) /

.( * ( * /1 3

Therefore,

( ) ∑

.0 .( ) ( ) /

.( ) ( ) /1

∑ 0 .( ) ( ) /

.( ) ( ) /1

5.4.2.3. The third moment of the truncated Weibull Lomax distribution

Setting r = 3 in (5.13), the third non-central moment of X may be achieved as follows:

∑

0 .( ) ( ) /

.( ) ( ) /1

Therefore,

∑

0 .( ) ( ) /

.( ) ( ) /1

∑

[ (( * ( * +

(( * ( * +]

∑ [ (( * ( * +

(( * ( * +]

5.4.2.4. The Forth moment of the truncated Weibull Lomax distribution

Finally, setting r = 4 in (5.10), the fourth non-central moment of X may be achieved as follows:

= ∑

0 .( ) ( ) / .( ) ( ) /1

Therefore,

∑

0 .( ) ( ) / .( ) ( ) /1

∑

0 .( ) ( ) / .( ) ( ) /1

∑

0 .( ) ( ) / .( ) ( ) /1

∑

0 .( ) ( ) / .( ) ( ) /1

5.4.2.5. The Variance of the truncated Weibull Lomax distribution

The variance can be calculated from the ordinary moments using well- known relationship:

∑

.0 .( ) ( ) / .( ) ( ) /1

∑ 0 .( ) ( ) /

.( ) ( ) /1

{ ∑ 0 .( ) ( ) /

.( ) ( ) /1 3

∑ 0 .( ) ( ) /

.( ) ( ) /1

∑ 0 .( ) ( ) /

.( ) ( ) /1

∑ 0 .( ) ( ) /

.( ) ( ) /1

∑ 0 .( ) ( ) /

.( ) ( ) /1

∑ 0 .( ) ( ) /

.( ) ( ) /1

∑ 0 .( ) ( ) /

(( * ( * +]

5.4.2.6. The Skewness of the truncated Weibull Lomax distribution

The Skewness measure can be computed from the ordinary moments using well-known relationship:

⁄ ( )

2 ∑ 0 .( ) ( ) /

.( ) ( ) /1

∑ 0 .( ) ( ) /

.( ) ( ) /1

∑ 0 .( ) ( ) /

.( ) ( ) /1 3⁄

{ ∑ 0 .( ) ( ) /

.( ) ( ) /1

∑ 0 .( ) ( ) /

⁄

.( ) ( ) /1 3

5.4.2.7. The Kurtosis of the truncated Weibull Lomax distribution

Analogues to the skewness measure, the kurtosis measure of the T-WL distribution may be computed using the ordinary moments through the well-known relationship:

( )

where can be computed using Eq. (5.17) and can be computed using the form Eq. (5.15) above.

5.4.3. Coefficient of Variation

Coefficient of Variation (CV) is expressed as the ratio of standard deviation and mean. The CV is a measure of variability in the data. When the value of the CV is high, it means that the data has high variability and low stability. When the value of coefficient of variation is low, it means the data has low variability and high stability. The formula for the CV of T-WL distribution is given below in :

√ ⁄

where can be computed using Eq. (5.14), can be computed using Eq. (5.15) and Var(X) can be computed using Eq. (5.18) above.

5.4.4. The Mode of the truncated Weibull Lomax ( ) distribution

The mode of the truncated Weibull Lomax ( ) distribution is the value x at which the pdf has its maximum value. This value may be obtained by maximizing the pdf function of the truncated Weibull Lomax ( ) distribution as follows:

(( ) * [ ( ) (( ) ) ]

Thus,

(( ) ) 2 0 [( * ] (( * *

(( ) ) ( * [(( * * ]

(( ) * ( ) (( ) ) [ ]}⁄

Therefore,

(( ) ) 2 0 ( ) ( ) (( ) )

(( ) * ( ) (( ) ) *( ) +

(( ) * 0 (( ) ) 1 ( ) (( ) ) }⁄

And after some steps,

.( * / ( ) (( ) * 0 ( ) (( ) * ( ) 1

(( ) * ( ) (( ) ) [ ( ) (( ) )

( ) ] .

This implies that either:

(( ) * ( ) , (( ) ) , or

[ ( ) (( ) ) ( ) ] =0.

Since , then all roots of ( ) (( ) )

(( ) * are rejected, but we have

[ ( ) (( ) ) ( ) ]

Then, ( ) (( ) ) ( )

Thus, we have:

( ) [ (( ) ) ]

Therefore, we get the mode of the truncated Weibull Lomax ( ) distribution numerically at (5.22). It is exactly the same as the mode the Weibull Lomax ( ) distribution.

5.4.5. Incomplete moments

Theorem 5.1. th The r incomplete moment, say ∫ , of X which follows the truncated Weibull Lomax ( ) distribution is given by substituting r=1, r=2,...... , in the following equation,

∑

* ( ) (( ) )+

Proof: th To obtain the r incomplete moment, say ∫ , of low order of r, such as r=1, r=2,...... ,for X which follows the truncated Weibull Lomax ( ) distribution, it is much easier to first obtain as follows:

∫

Since from then

∫

∑ ∫

∑ ∫ Since,

is the Exponentiated Lomax distribution (Tahir, et al. (2010)),

∑ ∫ ( ) ( ) (

( ) )

∑ ∫ . ( * /

∑ ∫ ( ) ( )

( ( ) )

∑ ∫ .( * / ( *

. ( * /

∑ ∫ (( ) )

( ( ) )

Let ( ) then ,

( ) .

( ) ( ) ∑ ∫

∑

( ) ( ) ( ) 0 1 ∫ ∫

∑

( ) ( ) ( ) 0 1 ∫ ∫

Using the complete and incomplete beta functions :

∫ ,

∫

Then (5.23) implies that

∑

* ( ) (( ) )+

The first incomplete moments of the truncated

Weibull-Lomax distribution where obtain from of low order of r, such as r=1, r=2,...... , it is much easier to obtain it by substitute it in .

For Ex., at r=1, we have ∑

* ( ) (( ) )+

then,

∑

[ ( * (( ) *]

Incomplete moments can be used to obtain Mean deviations and Bonferroni and Lorenz curves which has many applications in economics, reliability, demography, insurance and medicine.

5.4.6. Bonferroni and Lorenz Curves

Bonferroni and Lorenz curves have applications not only in economics to study income and poverty, but also in other fields like reliability, demography, insurance and medicine. They are defined by

∫ and

∫

respectively, where and .

From∫ , or ∫ we have

, and , (Paranaiba, et al., 2010).

5.4.7. Moment Generating Function

Let be a random variable following the truncated distribution. The of X, say [ ] may be obtained by computing the mgf of as follows,

( )

∫

∫ ∑

∑ ∫

∑

where follows from Eq. above. Thus,

∑ ∑

0 .( ) ( ) / .( ) ( ) /1

∑

0 .( ) ( ) / .( ) ( ) /1

Now, since is the of X for the distribution and is defined by ( ) , the formula of the mgf of the distribution follows below,

( )

∑

0 .( ) ( ) / .( ) ( ) /1

5.4.8. Mean deviations

The mean deviations about the mean can be used as a measure of spread or scatter of the data from the median as the center of the population. Let and be the mean and the median of the distribution, respectively. Following Ashour & Eltehiwy (2013), the mean deviations about the mean and about the median may be computed as follows:

∫ , and ∫ respectively. From equations in chapter (3) we have:

∫ , and

∫ that, where denotes the first incomplete moment calculated from by substituting r =1, where:

∑

* (( ) )+

which yields:

∑ * ( )

(( ∑

* (( ) ( ) * (( ) ( ) *+ )

And from , the cdf of for T- WL distribution we have,

(( ) * (( ) * [ ] which yields

.( * / [

0 (( ∑ * (( ) ( ) *

(( ) ( ) *+) * 1

Substituting equations , we conclude the mean deviations about the mean . Similarly, Substituting equations

we conclude the mean deviations about the median .

5.4.9. Mean residual life

The MRL function of a variable from computes the expected remaining survival time of a subject given survival up to time X. Assuming that and following in Poynor (2010),

∫ then the MRL function for continuous X is defined as:

∫ ∫ ⟨ ⟩

Now, since with survival function as in , the MRL function is defined as the expected value of the remaining lifetimes after a fixed time point x, i.e.

∫

.( * / .( * /

∫

[ ]

.( * / .( * /

( { })

This integral is indefinite but the mean residual life could be computed numerically.

5.4.10. The Quintile function of the truncated Weibull Lomax( ) distribution

The quantile of a random variable X is the smallest value such that

X has a probability p of assuming a value equal to or less than

[ ] [ ]

In terms of the distribution function F, the quantile function Q returns the value x such that

( ) ( )

Simulating T-WL random variable is straightforward. Let p be the uniform variable on the unit interval (0, 1]. Thus, using the inverse transformation method. By inverting (5.6), the quantile function (qf) of X follows as

.( * / .( * /

( ) 8 9

.( * / .( * /

.( * / .( * / 4 5 4 5

⁄

.( * / ( * 6 4 57

⁄ ⁄

.( * / { 6 4 57 }

Let

⁄ ⁄

.( * / { 6 4 57 }

Thus, if the random variable X in (5.30) has pdf given by , the variable follows the truncated .

Special quantiles:

In particular, from the median is given by:

⁄ ⁄

.( * / { 6 4 57 }

th A quantile of order ⁄ )the 50 percentile) is used as a measure of the center of the distribution. The first quartile and may be computed as follow,

⁄ ⁄

.( * / { 6 4 57 }

And, the third quartile and may be computed as follows,

⁄ ⁄

.( * / { 6 4 57 }

Assuming uniqueness, let denote the first, second, and third quartiles of X, respectively. Note that the interval from to , is the interquartile range and defined as: .

5.5. Summary

In the present chapter we derived the probability density function and the cumulative distribution function of the doubly truncated four-parameter Weibull- Lomax distribution. We also derived the Survival function, the Hazard rate function and the reversed-hazard rate function of the distribution. The moment, including the expectation, the second, the third and the forth moments as well as the variance were also obtained. Other important measures such as the Skewness, the Kurtosis, the Coefficient of Variation and the mode were derived. We showed that they could be computed numerically. The mode of the doubly truncated four- parameter Weibull-Lomax distribution is exactly the same as the mode the four- parameter Weibull Lomax distribution. The incomplete moment and its use in building the Bonferroni and Lorenz curves were presented. We also established the moment generating function, the mean deviations about the mean, the mean residual life, and the quantiles including the median, the first quartile and the third quartile.

In the next chapter, we are going to apply all the established results to two datasets using the R software and illustrate the flexibility of our distributions using the goodness of fit and some plots.

Chapter 6

Application of the Weibull-Lomax and Truncated Weibull- Lomax distributions to two real data sets

6.1. Introduction

In this chapter, the usefulness of the four-parameter WL distribution is illustrated. Two real data sets are used to compare the WL distribution's fit with other generalizations of Lomax distributions and Weibull distribution to illustrate the flexibility of the WL distribution. On the other hand the usefulness of the four- parameter Truncated WL distribution is illustrated. The two data sets are used to compare the T-WL distribution's fit with the other generalizations of Lomax distributions and Weibull distribution and our Original Weibull Lomax distribution to illustrate the flexibility of the T-WL distribution. The first data set is the lifetime of 242 Breast Cancer Patients' data in Gaza Strip, obtained from Okasha & Matter (2015) and the second data set is the failure times of 84 Aircraft Windshield, obtained from Blischke & Murthy (2000). The first data set has been claimed that the three-parameter Burr XII distribution provides the best fit for the data. In order to obtain a fair comparison, we use some goodness of fit criteria.

6.2. Possible alternative distributions

The fits of the WL distribution will be compared with the well know distributions: the Kumaraswamy-Generalized Lomax Distribution (KwGlx) (Shams, 2013), Exponential Lomax Distribution (ExplLx) (El-Bassiouny, et. al., 2015), the Exponentiated Lomax Poisson Distribution (ELP) (Ramos, et al., 2013), the Gumbel-Lomax Distribution (GuLx) (Tahir, et al., 2015), Exponentiated Lomax Distribution (ExpdLx) (Abdul-Moniem & Abdel-Hameed, 2012), Lomax

Lingappaiah (1987) and Weibull distribution (Weibull, 1951). On the other hand the fit of the T-WL distribution will be also compared with the distributions listed above in addition to the original WL distribution. The density functions of those distributions are given by:

Kumaraswamy-Generalized Lomax (KwGlx):

( * 2 ( * 3 0 2 ( * 3 1

0 2 ( * 3 1

Exponential Lomax (ExplLx):

( *

Exponentiated Lomax Poisson (ELP):

. ( * / ( ( ) *

( )

. ( * / 4 5

Gumbel-Lomax (GuLx) :

⁄

.( * / ( )

{ ( ) }

⁄

.( * /

Exponentiated Lomax (ExpdLx):

( * . ( * /

. ( * /

Lomax Distribution:

( *

111

Weibull Distribution:

( ) ( )

( )

To demonstrate which distribution gives a better fit for each data set, numerical evaluations are carried out using the R software. The following tables provide the MLEs of the distributions' parameters. The model selection is carried out using the Akaike information criterion (AIC), the Bayesian information criterion (BIC), the consistent Akaike information criterion (CAIC), the Hannan– Quinn information criterion (HQIC) and the Cramér-von Mises statistic ( ) , Anderson-Darling ( ) and where:

[ ],

Where denotes the log-likelihood function evaluated at the maximum likelihood estimates, is the number of parameters and is the sample size. For more details about those criteria see Chen & Balakrishnan (1995).

6.3. The Data and the distribution's fits

These data are used here only for illustrative purposes. The required numerical evaluations are carried out using the R software.

6.3.1. First data set: Lifetime of Breast Cancer Patient's Data in Gaza Strip

The first data set is the Lifetime of Breast Cancer Patient's Data in Gaza Strip. This data set has taken from Okasha & Matter (2015) where it has originally been obtained from the Ministry of Health in Gaza City, on a random sample of incidence dates and death dates of 1,000 breast cancer patients within a period of 5

111 years starting from the beginning of 2009 to end of 2013. The total study period in days was around 1820 day and by subtracting incidence date from death date we get the lifetime for breast cancer patients in the Gaza Strip in days. Since the number of observations was 1000 patients, including 703 patients were still alive and 55 patients were omitted because their history of the disease was such that the date of death is the same as the incidence dates, and the remaining 242 patients have life time shown in Table 6.1.

Table 6.1. Data of lifetime (days) of (242) breast cancer patients in Gaza Strip.

11, 73, 173, 283, 402, 516, 674, 779, 1142, 17, 82, 175, 284, 403, 520, 676, 784, 1147, 25, 89, 177, 290, 403, 530, 679, 786, 1156, 27, 89, 177, 292, 406, 532, 680, 795, 1164, 27, 89, 178, 293, 406, 537, 682, 822, 1212, 29, 90, 178, 294, 422, 538, 686, 830, 1231, 29, 104, 181, 298, 423, 543, 698, 831, 1238, 29, 106, 189, 300 , 434, 548, 699, 837, 1267, 29, 118, 190, 302, 439, 555, 716, 848, 1285, 29, 118, 196, 314, 444, 558, 716, 889, 1307, 30 , 118, 198, 323, 449, 562, 722, 898, 1332, 31, 118, 205, 323, 457, 574, 722, 906, 1361, 34, 139, 207, 325, 457, 578, 726, 912, 1364, 38, 145, 207, 326, 463, 583, 726, 916, 1368, 38, 145, 213, 334, 466, 591, 729, 933, 1380, 41, 145, 219, 337, 468, 593, 732, 962, 1412, 47, 146, 227, 339, 470, 597, 735, 992, 1416, 49, 147, 231, 347, 482, 607, 742, 995, 1456, 57, 148, 234, 347, 484, 609, 745, 1004, 1554, 57, 148, 234, 351, 487, 616, 749, 1029, 1597, 59, 148, 236, 354, 494, 616, 752, 1048, 1614, 59, 158, 236, 355, 498, 629, 752, 1052, 1756, 59, 160, 237, 362, 499, 646, 764, 1053, 1790, 59, 161, 251, 363, 499, 647, 764, 1069, 1813, 60, 161, 257, 367 ,502, 653, 770, 1091, 1835, 65, 164, 272, 367, 503, 663, 771, 1120, 1997, 66, 168 280, 379, 504, 668, 776, 1140.

In Table 6.2, we list the MLEs of the parameters and their corresponding standard errors (in parentheses) for all fitted distributions to the data. The statistics AIC, CAIC, BIC, HQIC, , and were computed and the results are listed in Table 6.3.

These results show that the Weibull-Lomax distribution has the lowest of all values of the statistics AIC, BIC, CAIC, HQIC, , and among all the other fitted distributions. This result confirms that the Weibull-Lomax distribution should be chosen as the best distribution for the data.

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Table 6.2. MLEs of the parameters and its SE's of different distributions for the first data set. Lomax Gumbel Dist. WL - - 6.3473 5.1458 0.0925 1.7652 - - (S.E) - - (4.319) (2.164) (0.017) (4.644) - - Exponentiated Lomax - - 8.4183 - 0.9187 20.660 - - (S.E) - - (0.149) - (0.024) (1.112) - - ELP -3.341 6.4747 - - 1.1798 20.137 - - (S.E) (0.593) (2.790) - - (0.089) (11.03) - - GuLx - - - - 8.7538 10.621 10.900 30.213 (S.E) - - - - (1.453) (0.407) (0.541) (8.651) KwGlx - - 6.0744 11.899 0.2995 13.447 - - (S.E) - - (0. 001) (3.971) (0.053) (0.647) - - Lomax - - - - 0.4364 38.778 - - (S.E) - - - - (0.035) (5.139) - - Weibull - - 35.656 0.3054 - - - - (S.E) - - (4.620) (0.018) - - - -

Table 6.3. Goodness of fit of different distributions for the first data set.

AIC CAIC BIC HQIC WL 3528.831 3529 3542.787 3534.453 1.639722 0.2622486 1760.416 KwGlx 3578.817 3578.985 3592.772 3584.439 5.949424 0.9686881 1785.408 ELP 3600.605 3600.774 3614.561 3606.227 7.571384 1.249792 1796.303 GuLx 3614.415 3614.583 3628.37 3620.036 8.331557 1.368601 1803.207 ExpdLx 3636.517 3636.618 3646.984 3640.734 10.74345 1.805012 1815.259 Lomax 3825.218 3825.268 3832.196 3828.029 11.29614 1.8978696 1910.609 Weibull 4095.844 4095.895 4102.822 4098.655 2.868221 0.4607552 2045.922

Additionally, in order to assess if the distribution's fit is appropriate, a histogram of the data with plots of the curves of the WL, KwGlx, ELP, GuLx, ExpdLx, Lomax and Weibull distributions superimposed is shown in Figure 6.1 (a) and (b). From these plots, we can conclude that the WL distribution yields the best fit and hence it can be described as the most adequate distribution to be fitted for the Lifetime of Breast Cancer Patient's Data in Gaza Strip.

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Figure 6.1. Histogram of the first data set and its fitted of pdfs.

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Since the values of the goodness of fit for the Weibull-Lomax distribution are smaller than those values of the other distributions, and the plots of the fitted histogram of the data and the superimposed curve of the Weibull-Lomax distribution is closer to the histogram than all those of other distributions, then we may conclude that the four-parameter Weibull-Lomax distribution is a very competitive distribution to this dataset.

6.3.1.1. Estimation of the parameters of Weibull Lomax ( ) distribution from the lifetime breast cancer patient's data in Gaza Strip

Assuming that the breast cancer patients in Gaza Strip data follow the Weibull Lomax ( ) distribution, then the estimates of four parameters of the distribution using the R software are:

̂ ̂ ̂ ̂

6.3.1.2. Expected lifetime of breast cancer patients from original lifetime data

Expected mean survival time of breast cancer patients based on these data and the four parameter Weibull Lomax distribution using the R software is 534.3649 day which is roughly a year and a half; and the variance and standard deviation are 231800.6 and 481.4567 respectively. The mean survival time of breast cancer patients seem to be high due to several reasons, including the type of cancer and benign or malignant nature of the disease and type of treatment for different classes of patients.

6.3.2. The second data set: Failure times of Aircraft Windshield

The windshield on a large aircraft is a complex piece of equipment, comprised basically of several layers of material, including a very strong outer skin with a heated layer just beneath it, all laminated under high temperature and pressure. Failures of these items are not structural failures. Instead, they typically involve damage or delimitation of the nonstructural outer ply or failure of the heating system. These failures do not result in damage to the aircraft but do result in replacement of the windshield.

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Data on all windshields are routinely collected and analyzed. At any specific point in time, these data will include failures to date of a particular distribution as well as service times of all items that have not failed. Data of this type are incomplete in that not all failure times have as yet been observed.

Data on failure and service times for a particular model windshield are given in Table 6.4 of Blischke & Murthy (2000). The data consist of 153 observations, of which 84 are classified as failed windshields, and the remaining 65 are service times of windshields that had not failed at the time of observation. The unit of measurement is 1000 hours.

Table 6.4. Data of failure times of (84) Aircraft Windshield 0.040, 1.866, 2.385, 3.443, 0.301, 1.876, 2.481, 3.467, 0.309, 1.899, 2.610, 3.478, 0.557, 1.911, 2.625, 3.578, 0.943, 1.912, 2.632, 3.595, 1.070, 1.914, 2.646, 3.699, 1.124, 1.981, 2.661, 3.779,1.248, 2.010, 2.688, 3.924, 1.281, 2.038, 2.82,3, 4.035, 1.281, 2.085, 2.890, 4.121, 1.303, 2.089, 2.902, 4.167, 1.432, 2.097, 2.934, 4.240, 1.480, 2.135, 2.962, 4.255, 1.505, 2.154, 2.964, 4.278, 1.506, 2.190, 3.000, 4.305, 1.568, 2.194, 3.103, 4.376, 1.615, 2.223, 3.114, 4.449, 1.619, 2.224, 3.117, 4.485, 1.652, 2.229, 3.166, 4.570, 1.652, 2.300, 3.344, 4.602, 1.757, 2.324, 3.376, 4.663.

In Table 6.5, we list the MLE estimates of the parameters for different fitted distributions and their corresponding standard errors (in parentheses).

Table 6.5. MLEs and SE's of different distributions for the second data set.

Lomax DIST WL - - 0.0118 0.6462 5.9950 1.3790 - - (S.E) - - (0.009) (0.593) (5.431) (1.008) - - Exponential - 0.0237 - - 6.3074 3.7412 - - Lomax - (0.052) - - (0.021) (0.136) - - (S.E) KwGlx - - 2.9739 19.877 2.2171 12.385 - - (S.E) - - (0.401) (26.13) (2.452) (8.674) - - GuLx - - - - 5.7000 3.7700 1.7e+5 5.1e+4 (S.E) - - - - (0.827) (0.335) (9219) (5700) Exponentiated - - 3.5417 - 11653 15521 - - Lomax - - (0.604) - (1180) (126.1) - - (S.E) ELP 1e-10 3.5473 - - 22063 29382 - - (S.E) (0.110) (1590) - - (1590) (271.9) - - Lomax - - - - 30087 76941 - - (S.E) - - - - (3294) (387.5) - -

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The estimated values of the statistics AIC , CAIC , BIC , HQIC , , and are listed in Table 6.6 below.

Table 6.6. Goodness of fit of collection of different distributions for the second data set.

AIC CAIC BIC HQIC WL 263.675 264.175 273.446 267.605 0.58555 0.08658 127.837 ExplLx 267.099 267.099 274.427 270.046 1.03596 0.16777 130.549 KwGlx 277.776 278.276 287.546 281.706 0.91460 0.09705 134.888 GuLx 278.143 278.643 287.913 282.073 0.88721 0.09486 135.071 ExpdLx 288.811 289.107 296.139 291.758 1.73236 0.21764 141.405 ELP 290.808 291.308 300.578 294.738 1.74460 0.21964 141.404 Lomax 333.978 334.124 338.863 335.943 1.99471 0.25060 164.989 These results show that the Weibull-Lomax distribution has the lowest criteria of all AIC, BIC, CAIC, HQIC, , and values for all the fitted distributions, and thus it should be chosen as the best distribution.

Additionally, in order to assess how the distribution is appropriate, the histogram of the data and the pdf curves of all the WL, ExplLx, KwGlx, GuLx, ExpdLx, ELP and Lomax distributions are superimposed and shown in Fig. 6.2 (a) and (b). From these plots, we can conclude that the WL distribution yields the best fit and hence confirms its adequacy to the data.

Since the values of the goodness of fit for the Weibull-Lomax distribution are smaller than those of the other distributions, and the plots of the fitted histogram of the data and the superimposed curve of the Weibull-Lomax distribution is closer to the histogram than those of all other distributions, then the Weibull-Lomax distribution is a very competitive distribution to these data.

6.3.2.1. Estimation of the parameters of Weibull Lomax ( ) distribution from the failure times of Aircraft Windshield data

Assuming the failure times of Aircraft Windshield data follow the Weibull Lomax ( ) distribution, then the estimates of four parameters using the R software are:

̂ ̂ ̂ ̂

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Fig. 6.2. Histogram of the second data set and its fitted of pdfs.

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6.3.2.2. Expected mean lifetime of Aircraft Windshield from original lifetime data

Expected mean lifetime of Aircraft Windshield using the four-parameter Weibull Lomax distribution and the R software is 2.555561 units which is 2555 hours; and the variance and standard deviation are 1.24336 and 1.11506 respectively. The mean survival time until failure of Aircraft Windshield data seem to be high due to several reasons, including the type of Windshield and high or low flight and type of detergent used.

Therefore, the WL distribution has been fitted to two real data sets and compared with other known distributions. The results show that the WL distribution gives the best fit to each data set.

6.4. The truncated Data and the distribution's fit

These data are used here only for illustrative purposes. The same data sets described in the previous section were used to illustrate the fit of the Truncated Weibull Lomax distribution and performance of the estimates of its parameters. The required numerical evaluations are carried out using the of R software.

6.4.1. First data set: Lifetime of Truncated Breast Cancer Patient's Data in Gaza Strip

The Lifetime of truncated Breast Cancer Patient's Data in Gaza Strip between 100 and 1000 days. This data set from the original (non-truncated) data has been discussed in section 6.3.1 above. The original data represents the lifetime of Breast Cancer patients measured in days for 242 patients. However, the data were truncated to fall between 100 to 1000 days. This is legitimate act since patients who have lifetime bellow 100 days are thought to be wrongly reported and patients who have lifetime above 1000 days are thought to be outliers or may be treated differently in a foreign country. The number of observations becomes 175 patients. The truncated lifetime data is shown in table 6.7 below. In Table 6.8, we list the MLEs of the parameters and SE's for all fitted distributions. and the AIC, CAIC, BIC, HQIC, , and statistics are listed in Table 6.9.

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Table 6.7. Truncated lifetime data (in days) of (175) breast cancer patients in Gaza Strip.

173, 283, 402, 516, 674, 779, 175, 284, 403, 520, 676, 784, 177, 290, 403, 530, 679, 786, 177, 292, 406, 532, 680, 795, 178, 293, 406, 537, 682, 822, 178, 294, 422, 538, 686, 830, 104, 181, 298, 423, 543, 698, 831, 106, 189, 300, 434, 548, 699, 837, 118, 190, 302, 439, 555, 716, 848, 118, 196, 314, 444, 558, 716, 889, 118, 198, 323, 449, 562, 722, 898, 118, 205, 323, 457, 574, 722, 906, 139, 207, 325, 457, 578, 726, 912, 145, 207, 326, 463, 583, 726, 916, 145, 213, 334, 366, 591, 729, 933, 145, 219, 337, 468, 593, 732, 972, 146, 227, 339, 470, 597, 735, 992, 147, 231, 347, 482, 607, 742, 995, 148, 234, 347, 484, 609, 745, 148, 234, 351, 487, 616, 749, 148, 236, 354, 494, 616, 752, 158, 236, 355, 498, 629, 752, 160, 234, 362, 499, 646, 764, 161, 251, 363, 499, 504, 379, 280, 168, 771, 663, 647, 764, 161, 257, 367, 502, 653, 770, 164, 272, 647, 503, 776, 668.

Table 6.8. MLEs of the parameters and its SE's of different distributions for the first truncated data set. Lomax Gumbel DIST T-WL - - 8.3653 4.8121 0.0969 4.2086 - - (S.E) - - ( 26.67) (2.010) ( 0.029) (7.030) - - WL - - 2.7184 6.5489 0.1380 5.8220 - - (S.E) - - (4.319) ( 0.960) (0.017) (4.825) - - Exponentiated Lomax - - 29.303 - 1.5839 39.042 - - (S.E) - - (6.072) - (0.115) (8.127) - - ELP 15.722 16.408 - - 0.5210 13.682 - - (S.E) (5.022) (2.168) - - (0.063) ( 3.305) - - GuLx - - - - 16.308 0.2922 0.3425 9.3387 (S.E) - - - - (11.84) (0.407) (0.541) (7.333) KwGlx - - 18.754 8.9106 0.6136 13.982 - - (S.E) - - (2.702) (3.743) (0.094) (3.650) - - Lomax - - - - 0.4155 36.951 - - (S.E) - - - - ( 0.039) (5.525) - - Weibull - - 32.541 0.3098 - - - - (S.E) - - (4.996) (0.023) - - - -

Table 6.9. Goodness of fit of different distributions for the first truncated data set.

AIC CAIC BIC HQIC T-WL 2355.94 2356.17 2368.57 2361.06 0.62155 0.09329 1173.97 WL 2389.12 2389.36 2401.76 2394.25 1.90650 0.27685 1190.56 KwGlx 2413.21 2413.44 2425.84 2418.33 3.45893 0.54312 1202.60 ELP 2413.81 2414.05 2426.45 2418.94 3.48531 0.54607 1202.90 ExpdLx 2442.39 2442.53 2451.86 2446.23 5.06074 0.83170 1218.19 GuLx 2451.35 2451.58 2463.98 2456.47 6.60257 1.12206 1221.67 Lomax 2787.54 2787.62 2793.85 2790.10 7.49931 1.27829 1391.77 Weibull 2993.28 2993.35 2999.60 2995.84 2.78484 0.42548 1494.64

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The above results show that the truncated Weibull-Lomax distribution has the lowest AIC, BIC, CAIC, HQIC, , and values among all the fitted distributions, which indicates that it should be chosen as the best distribution for the truncated data. Additionally, in order to assess if the distribution is appropriate, the histogram of the truncated data and plots of the pdf curves of the T-WL, WL, KwGlx, ELP, GuLx, ExpdLx, Lomax and Weibull distributions are superimposed and shown in Fig. 6.3(a) and (b). From those plots, we can conclude that the T-WL distribution yield the best fit and hence judged adequate for this truncated data set.

Since the goodness of fit statistics are very small for the T-WL distribution compared with those of the original WL and the other distributions, and the plots of the fitted histogram of the data and the superimposed curve of the T-WL distribution is closer to the histogram than all those of other distributions, then the T-WL distribution is a very competitive distribution to these truncated data and judged the most adequate distribution to the data.

6.4.1.1. Estimation of the parameters of Truncated Weibull Lomax ( ) distribution from the truncated lifetime breast cancer patient's data

Assuming that the truncated breast cancer patient's data in Gaza Strip data follow the Truncated Weibull Lomax ( ) distribution then the estimates of four parameters using the R software are:

̂ ̂ ̂ ̂

6.4.1.2. Expected mean lifetime of breast cancer patients from the truncated lifetime data

Expected mean survival time of breast cancer patients based on the truncated data and the four-parameter Truncated Weibull Lomax distribution using the R software is 462.134 days which is roughly a year and three months; and the variance and standard deviation are 64725.39 and 254.4118511 respectively. The mean survival time of breast cancer patients seem to be high due to several reasons, including the type of cancer and benign or malignant nature of the disease and the type of treatment the patients received.

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Figure 6.3. Histogram of the first truncated data set and its fitted of pdfs superimposed.

112

6.4.2. The second data set: Failure times of (67) Truncated Aircraft Windshield data

Data on failure and service times for a particular distribution of aircraft windshield are given in section 6.3.2 above. We truncated this data from 1 to 4. The data became only 67 instead of 84 observations. The truncated data are given in Table 6.10 below. In Table 6.11, we list the MLEs of the estimates of the parameters and their corresponding standard errors (in parentheses) for all fitted distributions to the Truncated data.

Table. 6.10. Data of failure times of (67) Truncated Aircraft Windshield.

1.866, 2.385, 3.443, 1.876, 2.481, 3.467, 1.899, 2.610, 3.478, 1.911, 2.625, 3.578, 1.912, 2.632, 3.595, 1.070, 1.914, 2.646, 3.699, 1.124, 1.981, 2.661, 3.779,1.248, 2.010, 2.688, 3.924, 1.281, 2.038, 2.82,3, 3.000, 1.281, 2.085, 2.890, 1.303, 2.089, 2.902, 1.432, 2.097, 2.934, 1.480, 2.135, 2.962, 1.505, 2.154, 2.964, 1.506, 2.190, 3.000, 1.568, 2.194, 3.103, 1.615, 2.223, 3.114, 1.619, 2.224, 3.117, 1.652, 2.229, 3.166, 1.652, 2.300, 3.344, 1.757, 2.324, 3.376.

Table 6.11. MLEs and its SE's of collection of different distributions for the second Truncated data set. Lomax DIST T-WL 5.7102 3.1367 0.6102 2.3940 (S.E) (11.35) (0.698) (0.317) (2.755) WL - - 0.0085 1.2743 6.0239 3.1025 - - (S.E) - - (0.008) (0.213) (0.283) (1.008) - -

Exponential Lomax - 0.0034 - - 5.5499 1.4821 - - (S.E) - (0.001) - - (1.222) (0.559) - - KwGlx - - 6.9616 10.643 7.9602 15.431 - - (S.E) - - (1.918) (14.31) (2.452) (14.57) - - GuLx - - - - 6.3914 0.5535 6.2661 8.2019 (S.E) - - - - (7.733) (0.210) (6.073) (5.66) Exponentiated - - 19.263 - 20.825 12.867 - - Lomax - - (5.697) - (12.09) (8.431) - - (S.E) ELP - 8.992 4.6786 - - 15.359 7.3969 - - (S.E) (11.5) (9.022) - - (11.47) (7.352) - - Lomax - - - - 66957 159034 - - (S.E) - - - - (7504) (149.7) - -

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The estimated values of the AIC , CAIC , BIC , HQIC , , and measures are listed in Table 6.12 below.

Table 6.12. Goodness of fit of collection of different distributions for the second truncated data set.

AIC CAIC BIC HQIC T-WL 147.454 148.099 156.273 150.944 0.21230 0.03979 69.7271 WL 156.539 157.185 165.358 160.029 0.58956 0.10036 74.2697 ExplLx 160.356 160.737 166.971 162.974 1.01677 0.17244 77.1782 KwGlx 155.873 156.518 164.692 159.363 0.44862 0.07014 73.9366 GuLx 161.244 161.889 170.062 164.733 0.77990 0.10972 76.6218 ExpdLx 159.036 159.417 165.650 161.654 0.60361 0.08405 76.5181 ELP 160.142 160.787 168.961 163.631 0.62305 0.08749 76.0667 Lomax 253.912 254.100 258.322 255.657 0.97345 0.14372 124.956

These results show that the truncated Weibull-Lomax distribution has the lowest AIC, BIC, CAIC, HQIC, , and values among all the other fitted distributions, and so it should be chosen as the best distribution for the data.

Additionally, in order to assess how appropriate the distribution is, the histogram of the data and plots of the pdf curves of the T-WL, WL, ExplLx, KwGlx, GuLx, ExpdLx, ELP and Lomax distributions are superimposed and shown in Fig. 6.4 (a) and (b). From these plots, we can conclude that the T-WL distribution yields the best fit and hence confirms its adequacy for the data.

Since the values of the goodness of fit of the truncated Weibull-Lomax distribution are smaller than those of the other distributions, and plots of the T-WL pdf curve and the histogram represents the data quite good, then the truncated Weibull-Lomax distribution is a very competitive distribution to these data.

6.4.2.1. Estimation of the parameters of Truncated Weibull Lomax ( ) distribution from the failure times of Truncated Aircraft Windshield data

Assuming that the failure times of Aircraft Windshield data follow the Truncated Weibull Lomax ( ) distribution then the estimates of four parameters using the R software are:

̂ ̂ ̂ ̂

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Figure 6.4: Histogram of the truncated second data set and its fitted of pdfs

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6.4.2.2. Expected mean lifetime of Aircraft Windshield from truncated lifetime data

Expected mean lifetime of Aircraft Windshield using the four-parameter Truncated Weibull-Lomax distribution is 2.365641 units which is 2366 hours; and the variance and standard deviation are 0.5418383 and 0.736096665 respectively. The expected mean lifetime of Aircraft Windshield seem to be high due to several reasons, including the type of Windshield and high or low flight and type of detergent used.

Therefore, the two real data sets were fitted to the T-WL and compared with WL and other known distributions. The results showed that the T-WL gives a good fit to each truncated data sets.

6.5. Summary

Based on all the foregoing illustrations and the goodness of fit on a collection of different important lifetime distributions, including the Weibull- Lomax distributions, in applied statistics, especially in survival analysis we may conclude the following: Breast cancer patient's data in Gaza Strip follow successfully the Weibull Lomax distribution. The expected mean survival time of breast cancer patients based on the data and the four-parameter Weibull Lomax distribution is 534.4 days. The expected mean survival time of breast cancer patients in Gaza Strip based on the truncated data and four-parameter truncated Weibull-Lomax distribution is 462.134 days. As for the second data set (Aircraft Windshield), the Aircraft Windshield data follow successfully the Weibull Lomax distribution. The expected mean lifetime of Aircraft Windshield based on the data and four-parameter WL distribution is 2.555561 units. The expected mean lifetime of Aircraft Windshield based on the truncated data and four parameter truncated Weibull Lomax distribution is 2.365641 units (one unit equals 1000 hours). As a result from criteria and plots the T-WL better than WL for both data sets.

In the next chapter, we give the final conclusion that concludes this study on the Weibull Lomax distribution, and Truncated Weibull Lomax distribution, and we suggest some recommendations for future studies.

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Chapter 7

Conclusion and Recommendations

7.1. Introduction

In this study we defined the four-parameter Weibull-Lomax distribution and made some inference on this distribution. Also we defined the doubly truncated four-parameter Weibull-Lomax with four parameters and made some inference on this modified version of Weibull-Lomax distribution. We concentrated on these distributions due to their importance in survival data, life testing and other fields (as is the case of the original distributions, Weibull and Lomax being important in life testing). We applied the results obtained theoretically on the Weibull-Lomax and doubly truncated Weibull-Lomax distributions throughout the thesis on two real datasets on lifetime of breast cancer patients in Gaza Strip and failure time of Aircraft Windshield. Finally, the following conclusions and recommendations may be stated.

7.2. Conclusions

From the results of this study, we can draw the following conclusions:

 The probability density function and the cumulative distribution function of the Weibull-Lomax distribution could be expressed in closed form.  The Weibull Lomax ( ) distribution could be generated through some transformations including the relation between the WL distribution and the well-known Weibull and exponential distributions.  The WL density function could be written as a double mixture of Exponentiated Lomax densities.  Various statistical properties for Weibull Lomax distribution including Hazard rate function, the quantile function, moments, coefficient of variation, Incomplete moments, Bonferroni and Lorenz curves, mean deviations, mean residual life, Entropies and moment generating function are investigated.

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 Various shapes of the density function and the hazard rate function, for various and fixed value of parameters are presented and discussed.  The maximum likelihood estimators of the parameters of the Weibull Lomax ( ) distribution is obtained numerically using Newton's methods since no closed form for the estimates of the parameters could be achieved.  Elements of information matrix were obtained and confidence intervals could be constructed for the parameters of the distribution from a sample of observations.  A simulation study concerning the estimators of the parameters of the Weibull Lomax ( ) distribution was conducted and presented in this thesis. We noted that the parameters' estimates are consistent since the standard error decreases as the sample size increases.  The probability density function and the cumulative distribution function of the doubly truncated Weibull Lomax ( ) are obtained in a closed form.  Special cases of the doubly truncated Weibull Lomax ( ), the right and the left truncated Weibull Lomax ( ) were obtained in closed forms.  Various properties of truncated Weibull Lomax ( ) including Hazard rate function, the Quantile function, Moments, Coefficient of Variation, Incomplete moments, Bonferroni and Lorenz Curves, Mean deviations, Mean residual life time and moment generating function are investigated.  Using various mathematical and numerical methods in R package, we obtained the parameters' estimates and the standard errors of the Weibull Lomax distribution and other different distributions for each of the Breast Cancer Patients' data and Aircraft Windshield data.  Using various mathematical and numerical methods in R package, we obtained the parameters' estimates and the standard errors of the Truncated Weibull

Lomax distribution and the other different distributions for each of the

Truncated Breast Cancer Patients' data and Truncated Aircraft Windshield data.  The mean survival time for breast cancer patients in Gaza Strip based on the original data is 534 days, while based on truncated data the mean survival time is 462 days.

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 The mean survival time for Aircraft Windshield based on the original data is 2.556 units, while based on truncated data the mean survival time is 2.366 units (one unit equals one thousand hours).

7.3. Recommendations

The Weibull Lomax ( ) and truncated Weibull Lomax ( ) distributions proved to be important to use in many applications. Therefore, from this study we can recommend the following:

 The Weibull Lomax ( ) distribution should be applied in fitting survival and lifetime data and its applications should be developed in survival analysis and other fields.  More research should be conducted to investigate other properties of the Weibull Lomax ( ) distribution that has not been discussed in this study, such as the likelihood ratio and other hypothesis tests on the parameters of the distribution.  More research should be conducted on the applications of the mathematical results of this study on various fields, especially medical, economics, quality assurance, environmental and engineering applications.  More research should be conducted to further modify other distributions that could be useful for modeling lifetime data particularly the heavy tailed ones.  More research should be conducted to create other relationships between the Weibull Lomax distribution and other survival distributions should be investigated and compared.

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