A Study of the Skew-Symmetric Models

Since Azzalini (1985,1986) introduced the fundamental properties of the skew-normal distribution, there are many investigations about the skew distributions based on certain symmetric probability density functions. These classes of the skew-symmetric distributions include the original symmetric distribution and have some properties like the original one and yet is skew. In this thesis, we consider two topics of the skew-symmetric models. In Chapter 1, we study the quadratic forms of multivariate skew normal-symmetric distributions. Following the paper by Gupta and Chang (2003) we generalize a multivariate skew normal-symmetric distribution with p.d.f. of the form fZ(z) = 2φp(z;Ω)G(α ′z), where Ω > 0, α ∈ R, φp(z;Ω) is the p-dimensional normal p.d.f. with zero mean vector and correlation matrix Ω, and G is taken to be an absolutely continuous distribution function such that G′ is symmetric about 0. First we obtain the moment generating function of certain quadratic forms. It is interesting to find that the distributions of some quadratic forms are independent of G. Then the joint moment generating functions of a linear compound and a quadratic form, and two quadratic forms, and conditions for their independence are given. Finally we take G to be one of normal, Laplace, logistic or uniform distribution, and determine the distribution of a special quadratic form for each case. In Chapter 2, we study the generalized skew-Cauchy distributions. We investigate the generalized skew-symmetric distributions. Suppose Y is an absolutely continuous random variable symmetric about 0 with probability density function f and cumulative distribution function F . If a random variable X satisfies X d = Y , then X is said to have a generalized skew distribution of F (or f). The generalized skew-Cauchy (GSC) distribution are considered and special examples of GSC distribution are presented. Some of these examples are generated from generalized skew-normal or generalized skew-t distributions.