Concomitants of order statistics from multivariate elliptical distributions

In this paper, by considering a 2n  -dimensional elliptically contoured random vector (XT,YT)T=(X1,…,Xn,Y1,…,Yn)T(XT,YT)T=(X1,…,Xn,Y1,…,Yn)T, we derive the exact joint distribution of linear combinations of concomitants of order statistics arising from XX. Specifically, we establish a mixture representation for the distribution of the rth concomitant order statistic, and also for the joint distribution of the rth order statistic and its concomitant. We show that these distributions are indeed mixtures of multivariate unified skew-elliptical distributions. The two most important special cases of multivariate normal and multivariate t distributions are then discussed in detail. Finally, an application of the established results in an inferential problem is outlined.

► Joint distribution of linear combinations of concomitants from elliptical distributions is derived.
► Distribution of concomitant and joint distribution of order statistic and its concomitant are derived.
► They are shown to be mixtures of multivariate unified skew-elliptical distributions.