Extended Generalized Skew-Elliptical Distributions and their Moments

This paper constructs a family of multivariate distributions which extends the class of generalized skew-elliptical (GSE) distributions, introduced by Azzalini and Capitanio (J. Roy. Statist. Soc. Ser. B, 65, 367–389, 2003), and derives formulae for it’s characteristics; mean and covariance. The extended GSE family is particularly relevant whenever the need arises to model data by skewed distributions, as is the case in actuarial science, risk management and other branches of science. The paper also generalizes the skew-normal distribution in the sense of Azzalini and Dalla Valle (Biometrika, 83, 715–726, 1996). Furthermore, for estimation purposes, the maximum likelihood equations are derived. Finally, a numerical examples is provided which demonstrates that the extended GSE distribution offers a better fit in comparison to the GSE distribution.