Order statistics from trivariate normal and tν-distributions in terms of generalized skew-normal and skew-tν distributions

Next, we introduce a generalized skew-tν distribution, which is a special case of the unified multivariate skew-elliptical distribution presented by Arellano-Valle and Azzalini [2006. On the unification of families of skew-normal distributions. Scand. J. Statist. 33, 561-574] and is in fact a three-parameter generalization of Azzalini and Capitanio's [2003. Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution. J. Roy. Statist. Soc. Ser. B 65, 367-389] univariate skew-tν form. We then use the relationship between the generalized skew-normal and skew-tν distributions to discuss some properties of generalized skew-tν as well as distributions of order statistics from bivariate and trivariate tν distributions. We show that these distributions of order statistics are indeed mixtures of generalized skew-tν distributions, and then use this property to derive explicit expressions for means and variances of these order statistics.