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Study of Generalized Lomax Distribution and Change Point Problem STUDY OF GENERALIZED LOMAX DISTRIBUTION AND CHANGE POINT PROBLEM Amani Alghamdi A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY August 2018 Committee: Arjun K. Gupta, Committee Co-Chair Wei Ning, Committee Co-Chair Jane Chang, Graduate Faculty Representative John Chen Copyright c 2018 Amani Alghamdi All rights reserved iii ABSTRACT Arjun K. Gupta and Wei Ning, Committee Co-Chair Generalizations of univariate distributions are often of interest to serve for real life phenomena. These generalized distributions are very useful in many fields such as medicine, physics, engineer- ing and biology. Lomax distribution (Pareto-II) is one of the well known univariate distributions that is considered as an alternative to the exponential, gamma, and Weibull distributions for heavy tailed data. However, this distribution does not grant great flexibility in modeling data. In this dissertation, we introduce a generalization of the Lomax distribution called Rayleigh Lo- max (RL) distribution using the form obtained by El-Bassiouny et al. (2015). This distribution provides great fit in modeling wide range of real data sets. It is a very flexible distribution that is related to some of the useful univariate distributions such as exponential, Weibull and Rayleigh dis- tributions. Moreover, this new distribution can also be transformed to a lifetime distribution which is applicable in many situations. For example, we obtain the inverse estimation and confidence intervals in the case of progressively Type-II right censored situation. We also apply Schwartz information approach (SIC) and modified information approach (MIC) to detect the changes in parameters of the RL distribution. The performance of these approaches is studied through simu- lations and applications to real data sets. According to Aryal and Tsokos (2009), most of the real world phenomenon that we need to study are asymmetrical, and the normal model is not a good model for studying this type of dataset. Thus, skewed models are necessary for modeling and fitting asymmetrical datasets. Azzalini (1985) in- troduced the univariate skew normal distribution and his approach can be applied in any symmet- rical model. However, if the underlying (base) probability is not symmetric, we can not apply the Azzalini’s approach. This motivated the study for more flexible alternative. Shaw and Buckley (2007) introduced a quadratic rank transmutation map (QRTM) which can be applied in any (symmetric or asymmetric) distribution. Recently, many distributions have been suggested using the QRTM to derive the transmuted class (TC) of distributions. This provides great flexibility in performing real datasets. We extend our work in RL distribution to derive the iv transmuted Rayleigh Lomax (TR-RL) distribution using the QRTM. Mathematical and statistical properties, such as moment generating function, L-moment, probability weight moments are de- rived and studied. We also establish the relationship between the TR-RL , the RL, and other useful distributions to show that our proposed distribution includes them as special cases. TR-RL is fitted to a well known dataset, the goodness of fit test and the likelihood ratio test are presented to show how well the TR-RL fits the data. v To the memory of my parents who paved the way for me during their life- time. To my husband Khalid Alghamdi and my children Lujain, Loai and Wael for their unconditional love and support. I dedicate this work. vi ACKNOWLEDGMENTS First of all, I would like to express my deepest thanks to my advisors Dr. Arjun Gupta and Dr. Wei Ning for their guidance, support, detailed comments, patience and invaluable encourage- ment they offered throughout this research. Under their guidance, I successfully overcame many difficulties and learned a lot. I would especially like to thank Dr. John Chen for his unforgettable support, and acceptance of being a member in my dissertation committee. I am also thankful to Dr. Jane Chang for the time she spent on reviewing this dissertation. Deepest sense of gratitude to Dr. Craig Zirbel for his careful and precious guidance throughout my study. My thanks and appreciation also to the staff of Department of Mathematics and Statistics at BGSU for their help and support. I must thank all of my friends in the US for helping and understanding me during this journey. Special thanks go to my brothers and my sisters for encouraging and inspiring me to follow my dreams. I am also grateful to my parents in law, who supported me emotionally and believed in me. I owe thanks to a very special person, my husband, Khalid Alghamdi for all his love, continuing support and encouragement. This entire journey would not have been possible without his support. Last but not least, I wish to express my love to my children Lujain, Loai and Wael who have unavoidably missed my presence during this period. vii TABLE OF CONTENTS Page CHAPTER 1 LITERATURE REVIEW . 1 1.1 Introduction . 1 1.2 Methodology . 3 1.3 Dissertation Structure . 7 CHAPTER 2 RAYLEIGH LOMAX DISTRIBUTION . 8 2.1 Introduction . 8 2.2 The Rayleigh Lomax Distribution . 8 2.3 Distributional Properties . 10 2.3.1 Shapes of pdf . 10 2.3.2 Moments . 11 2.3.3 L-moments . 14 2.3.4 Order statistics . 18 2.3.5 Quantile function . 20 2.3.6 Probability weighted moments . 20 2.3.7 Moment generating function . 22 2.4 Estimation . 22 2.4.1 MLEs of parameters . 23 2.4.2 Asymptotic distribution . 24 2.4.3 Simulation . 25 2.5 Application . 25 2.6 Hazard rate function . 31 viii 2.7 Inference under progressively type-II right-censored sampling for transformed Rayleigh Lomax distribution . 33 2.7.1 Interval estimation of parameter α ...................... 38 2.7.2 Interval estimation of parameter σ ...................... 39 2.7.3 Inverse estimation of parameters α and σ . 39 2.7.4 Simulation study . 40 2.7.5 An Illustrative Example . 46 2.8 Discussion . 48 CHAPTER 3 TRANSMUTED RAYLEIGH LOMAX DISTRIBUTION . 49 3.1 Introduction . 49 3.2 The Transmuted Rayleigh Lomax distribution . 50 3.2.1 Rank transmutation . 50 3.2.2 The Transmuted Rayleigh Lomax distribution . 52 3.3 Distributional Properties . 53 3.3.1 Shape of pdf . 53 3.3.2 Moments . 55 3.3.3 L-moments . 57 3.3.4 Order statistics . 60 3.3.5 Quantile function . 62 3.3.6 Probability weighted moments . 66 3.3.7 Moment generating function . 67 3.4 Estimation . 68 3.4.1 MLEs of parameters . 68 3.4.2 Asymptotic distribution . 70 3.4.3 Simulation . 73 3.5 Application . 73 3.6 Discussion and Conclusions . 76 ix CHAPTER 4 AN INFORMATION APPROACH FOR THE CHANGE POINT PROB- LEM OF THE RAYLEIGH LOMAX DISTRIBUTION . 78 4.1 Introduction . 78 4.2 Literature Review of the Change Point Problem . 79 4.3 Methodology . 80 4.4 Simulation Study . 85 4.5 Change Point Analysis for British Coal Mining Disaster . 88 4.6 Change Point Analysis for IBM Stock Price . 92 4.7 Change Point Analysis for the Radius of Circular Indentations . 94 4.8 Conclusions . 96 BIBLIOGRAPHY . 97 APPENDIX A SELECTED R PROGRAMS . 104 x LIST OF FIGURES Figure Page 1.1 Probability density function of Lomax distribution for different values of shape parameter α and λ = 1................................. 2 1.2 Cumulative density function of Lomax distribution for different values of shape parameter α and λ = 1................................. 3 1.3 Probability density function of Pareto (IV) distribution for different values of the shape parameter α and inequality parameter γ = 1; 2................. 4 2.1 Probability density function of Rayleigh Lomax as α increases and decreases. 11 2.2 Probability density function of Rayleigh Lomax as λ increases and decreases. 11 2.3 Probability density function of Rayleigh Lomax as σ increases and decreases. 12 2.4 Plot of the estimated densities for the Aircraft Windshield data. 28 2.5 Model of Probability density function of Rayleigh Lomax for the Aircraft Wind- shield data using the L-moments, the MLE and the method of moments. 29 2 1 2.6 Probability density function of the transformed Rayleigh Lomax for σ = 2 : . 31 2.7 Hazard rate function of the transformed Rayleigh Lomax for a random variable T when σ2 = 1=2..................................... 33 3.1 Probability density function of the TR-RL as α increases and decreases. 53 3.2 Probability density function of the TR-RL as λ increases and decreases. 54 3.3 Probability density function of the TR-RL as σ increases and decreases. 54 3.4 Probability density functions of TR-RL and RL (dot curve). 55 3.5 The relationship between α and the median of the TR-RL distribution. 64 3.6 The relationship between λ and the median of the TR-RL distribution. 65 xi 3.7 Fitted density curves to the remission time real data. 76 2 4.1 χ3 Q-Q plot of Sn as n=100, p-value=0.4376. 87 2 4.2 χ3 Q-Q plot of Sn as n=200, p-value=0.6342. 88 2 4.3 χ3 Q-Q plot of Sn as n=400, p-value=0.8649. 88 4.4 The auto-correlation plot of the British Coal Mining Disaster data. 91 4.5 Scatter plot for the British Coal Mining Disaster data. 91 4.6 SIC(k) values in the British Coal Mining Disaster data. 91 4.7 The auto-correlation plot of the transformed IBM data. 92 4.8 The IBM stock daily closing prices from May 17 of 1961 to November 2 of 1962. 93 4.9 The IBM stock daily closing prices rate from May 17 of 1961 to November 2 of 1962.
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