In this work 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 2 ¼ d Y 2 , 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. r 2007 Elsevier B.V. All rights reserved.MSC: primary 60E05; secondary 62E10
Following the paper by Gupta and Chang (Multivariate skew-symmetric distributions. Appl. Math. Lett. 16, 643-646 2003.) we generate a multivariate skew normal-symmetric distribution with probability density function of the form f Z ðzÞ ¼ 2f p ðz; XÞGða 0 zÞ, where X40; a 2 R p , f p ðz; XÞ is the p-dimensional normal p.d.f. with zero mean vector and correlation matrix X, and G is taken to be an absolutely continuous function such that G 0 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. r 2005 Elsevier B.V. All rights reserved.MSC: primary 62H10; secondary 62E15
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