Order statistics from trivariate normal and [formula omitted]-distributions in terms of generalized skew-normal and skew- [formula omitted] distributions

Abstract

We consider here a generalization of the skew-normal distribution, GSN ( λ 1 , λ 2 , ρ ) , defined through a standard bivariate normal distribution with correlation ρ , which is a special case of the unified multivariate skew-normal distribution studied recently by Arellano-Valle and Azzalini [2006. On the unification of families of skew-normal distributions. Scand. J. Statist. 33, 561–574]. We then present some simple and useful properties of this distribution and also derive its moment generating function in an explicit form. Next, we show that distributions of order statistics from the trivariate normal distribution are mixtures of these generalized skew-normal distributions; thence, using the established properties of the generalized skew-normal distribution, we derive the moment generating functions of order statistics, and also present expressions for means and variances of these order statistics. Next, we introduce a generalized skew- t ν 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- t ν form. We then use the relationship between the generalized skew-normal and skew- t ν distributions to discuss some properties of generalized skew- t ν as well as distributions of order statistics from bivariate and trivariate t ν distributions. We show that these distributions of order statistics are indeed mixtures of generalized skew- t ν distributions, and then use this property to derive explicit expressions for means and variances of these order statistics.