Multivariate Order Statistics Induced by Ordering Linear Combinations of Components of Multivariate Elliptical Random Vectors

In this chapter, by considering a np-dimensional random vector X1⊤,…,Xn⊤⊤$$\left ( {\mathbf {X}}_{1}^\top ,\ldots ,{\mathbf {X}}_{n}^\top \right ) ^\top $$, Xi∈ℝp$${\mathbf {X}}_{i} \in \mathbb {R}^{p}$$, i = 1, …, n, having a multivariate elliptical distribution, we derive the exact distribution of multivariate order statistics induced by ordering linear combinations of the components. These induced multivariate order statistics are often referred to as the concomitant vectors corresponding to order statistics from multivariate elliptical distribution. Specifically, we derive a mixture representation for the distribution of the rth concomitant vector, and also for the joint distribution of the rth order statistic and its concomitant vector. We show that these distributions are indeed mixtures of multivariate unified skew-elliptical distributions. The important special cases of multivariate normal and multivariate student-t distributions are discussed in detail. Finally, the usefulness of the established results is illustrated with a real dataset.