Multivariate Asymmetric Distributions of Copula Related Random Variables

Abstract

It is known that normal distribution plays an important role in analysing symmetric data. However, this symmetric assumption may not hold in many real word and in such cases, asymmetric distribution, including skew normal distribution, are known as the best alternative. Constructing asymmetric distributions is carried out using the conditional/selection approach of several independent variable conditioning on other set of variables and this approach does not work well when the independence between variables
violated. In this work we construct an asymmetric distribution when variables are dependent using a copula. Specifically, we consider the random vectors X and Y are connected using a copula function C_X,Y and we study the selection distribution Z = (X|Y ∈ T ).
We present some special cases of our proposed distribution, among them, multivariate skew-normal distribution. Some properties such as moments and moment generating function are investigated. Also, numerical analysis including simulation study as well as
a real data set analysis are presented for illustration.