Mathematics Ph.D. Dissertations

Statistical Inference for a New Class of Skew t Distribution and Its Related Properties

Date of Award

2017

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Statistics

First Advisor

Wei Ning (Committee Co-Chair)

Second Advisor

Arjun Gupta (Committee Co-Chair)

Third Advisor

Joseph Chao (Other)

Fourth Advisor

Junfeng Shang (Committee Member)

Abstract

Generalized skew distributions have been widely studied in statistics and numerous authors have developed various classes of these distributions, Cordeiro and de Castro (2011). To provide a wide and flexible family of modeling data that account for skewness and heavy tail weight in data automatically, Jones (2004) introduced the beta generated distribution as a generalization of the distribution of order statistics of a random sample from distribution F or by applying the inverse probability integral transformation to the beta distribution. Cordeiro and de Castro (2011) proposed a new class of distribution called the Kumaraswamy generalized distribution (KwF ) which is capable of fitting skewed data that cannot be fitted well by existing distributions. Azzalini (1985) introduced the univariate skew normal distribution as an extension of the normal distribution to accommodate asymmetry. Inspired by Azzalini’s work, numerous papers have been published on the applications of skewed distributions. Among all skewed distribution, the skew t distribution received special attention after the introduction of the skew multivariate normal distribution by Azzalini and Dalla Valle (1996). In this study we introduce new generalizations of the skew t distribution based on the beta generalized distribution and based on the Kumaraswamy generalized distribution. The new classes of distributions which we call the beta skew t (BST ) and the Kumaraswamy skew t distribution (KwST ) have the ability of fitting skewed and heavy tailed data and they are more general than the skew t distribution as they contain the skew t and some other important distributions as special cases. Related properties of the new distributions such as mathematical properties, moments, and order statistics are derived. A new approach of statistical inference using the L-moment as well as the classical maximum likelihood inference are used to estimate the parameters of the proposed distributions. The proposed distributions are applied to simulated data and real data to illustrate the fitting procedure. Further, we study the problem of analyzing a mixture of KwST distributions from the likelihood-based perspective. Computational technique using EM-type algorithm is employed for computing the maximum likelihood estimates. The proposed methodology is illustrated by analyzing simulated and real data examples.

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