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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. I الملخص يعتبر توزيع ويبل أحد التوزيعات الرياضية اﻷكثر استخداماً في نمذجة بيانات الحياة ، و لنمذجة الظواىر ذات معدل مخاطرة مطرد، يعتبر توزيع ويبل الخيار اﻷساسي وذلك بسبب أن شكل دالة كثافة اﻻحتمال ليذا التوزيع ينحرف يمنةً أو يسرةً . إﻻ أنو وبالرغم من كل ذلك فإنو ﻻ يممك اﻷسباب الكافية لنمذجة الظواىر الغير مطردة مثل الظواىر ذات الشكل اﻷشبو بحوض اﻻستحمام وحوض اﻻستحمام المقموب، التي تعتبر شائعةً في كل من د ارسة الموثوقية والبقاء. لذلك يعتبر ويبل من التوزيعات اﻷكثر مرونة في نمذجة بيانات الحياة ذات الطابع المطرد في معدل المخاطرة ولكنو غير عممي في نمذجة الظواىر الغير مطردة. في ىذه االدرسة نناقش تعميم جديد لتوزيع ويبل وىو توزيع ويبل لومكس ذو اﻷربع 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 اﻻستحمام أو حوض اﻻستحمام المقموب، وتمت د ارسة بعض الخواص والمعادﻻت الرياضية المرتبطة بالتوزيع الجديد. وقد تمكنا من اظيار أن توزيع ويبل لومكس يحتوي حاﻻت خاصة وىي التوزيع اﻷسي و توزيع ويبل . كما تمكنا من صياغة دالة كثافة اﻻحتمال لتوزيع ويبل لومكس كتوزيع مركب من كثافة احتمال توزيع لومكس اﻷسي ، وتمت مناقشة شكل كل من دالة كثافة اﻻحتمال ومعدل المخاطرة لمتوزيع الجديد، وقدرنا المعالم اﻷربعة لو بواسطة دالة اﻷرجحية العظمى، عﻻوة عمى ذلك، قمنا بتقديم ومناقشة توزيع ويبل لومكس المبتور بحالتيو الخاصتين لجية البتر، وادرسة بعض الخواص والمعادﻻت 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، واﻷخرى لفترة بقاء الزجاج اﻷمامي لمطائرات. وأظيرت النتائج بأن توزيع ويبل لومكس كان أفضل نموذج في ظل النماذج اﻷخرى المقارنة معو وذلك بنا ًء عمى سبعة من معايير جودة التوفيق اﻻحصائية في كﻻ من مجموعتي البيانات. وعند تطبيق توزيع ويبل لومكس المبتور بدا أنو أفضل من توزيع ويبل لومكس اﻷصمي. II 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 IV 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 V 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 VI 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.
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