On Bivariate Order Statistics from Elliptical Distributions

Abstract

In this article, by considering a (m + n)-dimensional random vector (XT, YT)T = (X1, …, Xm, Y1, …, Yn)T having a multivariate elliptical distribution, and denoting X(m) = (X(1), …, X(m))T and Y(n) = (Y(1), …, Y(n))T for the vectors of order statistics arising from X and Y, respectively, we derive the exact joint distribution of (X(r), Y(s))T, for r = 1, …, m and s = 1, …, n, and also joint distribution of (aTX(m), bTY(n))T where and . Further, by considering an elliptical distribution for the (m + n + 1)-dimensional random vector (X0, XT, YT)T, and treating X0 as a covariate variable, we present mixture representations for joint distributions of (X0, X(m), Y(n))T and (X0, aTX(m), bTY(n))T in terms of multivariate unified skew-elliptical distributions. These mixture representations enable us to obtain the best predictors of X0 based on X(m) and Y(n), and X0 based on aTX(m) and bTY(n), and so on. Finally, we illustrate the usefulness of our results by a real-life data.