Multivariate Likelihood Ratio Order for Skew-Symmetric Distributions with a Common Kernel

Werner Hürlimann

doi:10.1155/2013/614938

Abstract

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.

Highlights

According to Azzalini and Capitanio [1], the density function of the multivariate skew-symmetric distribution (SSD) with centrally symmetric density kernel f0(x), absolutely continuous univariate even density g0(x) =Gsk0󸀠e(wx)i,nganddismtruiblutitviaorniaGte0o(xd)d with an skewing weight w(x), is defined by f(x) = 2f0(x)G0{w(x)}
In the already large literature on skew-symmetric distributions, only a limited number of results establish in advance formal properties of the SSD by given qualitative properties of the kernel, skewing distribution and skewing weight, or equivalently by given kernel and perturbation function
If even transformation invariance holds between random vectors X and Y for all even T(x) that is, T(X)=dT(Y), the corresponding densities admit necessarily a representation with a common kernel

Summary

Introduction

According to Azzalini and Capitanio [1], the density function of the multivariate skew-symmetric distribution (SSD) with centrally symmetric (about 0) density kernel f0(x), absolutely continuous univariate even density g0(x) =. The SSD depends on the skewing distribution and the skewing weight only through the perturbation function G(x) = G0{w(x)} such that G(x) ≥ 0 and the reflective property G(x) + G(−x) = 1 holds. Any probability density function admits a uniquely defined SSD representation, as shown first by Wang et al. The present note considers the multivariate likelihood ratio order for multivariate skew symmetric distributions with a common kernel. Our analysis is restricted to SSDs with a common kernel This means that there exists a continuous centrally symmetric (about 0) density function f0(x), called kernel, and (reflective) perturbation functions GX(x) and GY(x) satisfying the conditions.

ISRN Probability and Statistics

Multiplication of these two inequalities yields

Discussion and Conclusions