On the correlation structures of multivariate skew-normal distribution

Abstract

Skew-normal distribution is an extension of the normal distribution where the symmetry of the normal distribution is distorted with an extra parameter. A multivariate skew-normal distribution has been parametrized differently to stress different aspects and constructions behind the distribution. There are several possible parametrizations available to define the skew-normal distribution. The current most common parametrization is through Ω and α, as an alternative, parametrization through Ω and δ can be used if straightforward relation to marginal distributions is of interest. The main problem with {Ω, δ}-parametrization is that the vector δ cannot be chosen independently of Ω. This motivated us to investigate what are the possibilities of choosing δ under different correlation structures of Ω. We also show how the assumptions on structure of δ and Ω affect the asymmetry parameter α and correlation matrix R of corresponding skew-normal random variable.