On the correlation structures of multivariate skew-normal distribution

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 Omega and alpha, as an alternative, parametrization through Omega and delta can be used if straightforward relation to marginal distributions is of interest. The main problem with {Omega,delta} parametrization is that the vector delta cannot be chosen independently of Omega. This motivated us to investigate what are the possibilities of choosing delta under different correlation structures of Omega. We also show how the assumptions on structure of delta and Omega affect the asymmetry parameter alpha and correlation matrix R of corresponding skew-normal random variable.