%0 Journal Article
%T Multivariate Likelihood Ratio Order for Skew-Symmetric Distributions with a Common Kernel
%A Werner H¨¹rlimann
%J ISRN Probability and Statistics
%D 2013
%R 10.1155/2013/614938
%X The multivariate likelihood ratio order comparison of skew-symmetric distributions with a common kernel is considered. Two multivariate likelihood ratio perturbation invariance properties are derived. 1. Introduction According to Azzalini and Capitanio [1], the density function of the multivariate skew-symmetric distribution (SSD) with centrally symmetric (about 0) density kernel , absolutely continuous univariate skewing distribution with an even density , and multivariate odd skewing weight , is defined by . The SSD depends on the skewing distribution and the skewing weight only through the perturbation function such that and the reflective property holds. Conversely, any function that satisfies these conditions ensures that is a density, which represents the SSD formulation adopted by Wang et al. [2]. In fact, any probability density function admits a uniquely defined SSD representation, as shown first by Wang et al. [2], Proposition . Azzalini and Regoli [3] refine this result to the representation of a density with arbitrary support in Proposition . The present note considers the multivariate likelihood ratio order for multivariate skew symmetric distributions with a common kernel. We obtain two general sufficient conditions in terms of a reverse hazard rate order (Theorem 4) and a weak reverse hazard rate order (Theorem 7) between perturbation functions. The second sufficient condition is related to Theorem 6.B.8 in Shaked and Shanthikumar [4], which establishes a sufficient condition for the stochastic order. It is simpler and implies even the likelihood ratio order. 2. Multivariate Likelihood Ratio Order Unless otherwise stated, and denote throughout real random vectors on some probability space with supports and . We assume that and have skew-symmetric distributions (SSD) in the sense of Azzalini and Capitanio [1] and Wang et al. [2] as unified in Azzalini and Regoli [3]. Our analysis is restricted to SSDs with a common kernel. This means that there exists a continuous centrally symmetric (about 0) density function , called kernel, and (reflective) perturbation functions and satisfying the conditions where the notation is used throughout, such that the probability density functions (pdf) of and are given by Equivalently to (2), there exists a continuous skewing distribution with an even density and odd skewing weights , such that Following Azzalini and Regoli [3, equation (9)], this equivalence is underpinned by the standard choice (made throughout) of a uniform random variable with distribution where denotes the indicator function of the set .
%U http://www.hindawi.com/journals/isrn.probability.statistics/2013/614938/