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)T, 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.