Mathematics Ph.D. Dissertations

Study of Generalized Lomax Distribution and Change Point Problem

Date of Award

2018

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics/Mathematical Statistics

First Advisor

Arjun Gupta (Committee Co-Chair)

Second Advisor

Wei Ning (Committee Co-Chair)

Third Advisor

John Chen (Committee Member)

Fourth Advisor

Jane Chang (Other)

Abstract

Generalizations of univariate distributions are often of interest to serve for real life phenomena. These generalized distributions are very useful in many ¿elds 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 ¿exibility 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 ¿t in modeling wide range of real data sets. It is a very ¿exible 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 con¿dence intervals in the case of progressively Type-II right censored situation. We also apply Schwartz information approach (SIC) and modi¿ed 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 ¿tting 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 ¿exible 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 ¿exibility in performing real datasets. We extend our work in RL distribution to derive the 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 ¿tted to a well known dataset, the goodness of ¿t test and the likelihood ratio test are presented to show how well the TR-RL ¿ts the data.

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