On Parametrization 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 called the shape parameter. The scalar version of the skew-normal distribution was introduced already in works by Azzalini (1985) and by Henze (1986). The multivariate skew-normal distribution has received considerable attention over the last years, but unfortunately there is no unique straightforward generalization from the scalar case. Therefore, various families of skew-symmetric distributions with different properties have been proposed and studied. In our work we refer to the “classical” multivariate skew-normal distribution introduced by Azzalini and Dalla Valle (1996). It must be noted that even the Azzalini’s skew-normal distribution can be parametrized in many different ways starting from the initial {Ψ,λ}$\lbrace {\bm\Psi},{\bm\lambda}\rbrace$-parametrization to the currently prevalent {Ω,α}$\lbrace {\bm\Omega},{\bm\alpha}\rbrace$-parametrization. This motivated us to search for viable alternatives and compare the overall behavior of different parametrizations, the key problems addressed in our article.