Exact distribution of a linear combination of a variable and order statistics from the other two variables of a trivariate elliptical random vector as a mixture of skew-elliptical distributions

We consider here a univariate skew-elliptical distribution, which is a special case of the unified multivariate skew-elliptical distribution studied recently by Arellano-Valle and Azzalini (2006) [1]. We then derive the exact distribution of a linear combination of a variable and order statistics from the other two variables in the case of a trivariate elliptical distribution. We show that the cumulative distribution function (cdf) of this linear combination is a mixture of the univariate skew-elliptical distribution functions.