Distribution of linear combinations of unordered and ordered components of an elliptical random vector

In this paper, by considering a (k + n)-dimensional random vector (XT, YT), X ∈ ℝk and Y ∈ ℝn, having a multivariate elliptical distribution, we derive the exact distribution of AX + LY(n), where A ∈ Rp×k, L ∈ ℝp×n, and Y(n) = Y(1), Y(2), …, Y(n))T denotes the vector of order statistics from Y. Next, we discuss the distribution of aTX+bY(r), for r = 1, …, n, a =(a1, …, ak)T ∈ ℝk and b ∈ ℝ. We show that these distributions can be expressed as mixtures of multivariate unified skew-elliptical distributions. Finally, we illustrate an application of the established results to stock fund evaluation.