Abstract
In this paper, by considering a (3n+1) -dimensional random vector (X0, XT, YT, ZT)T having a multivariate elliptical distribution, we derive the exact joint distribution of (X0, aTX(n), bTY[n], cTZ[n])T, where a, b, c∈ℝn, X(n)=(X(1), …, X(n))T, X(1)<···<X(n), is the vector of order statistics arising from X, and Y[n]=(Y[1], …, Y[n])T and Z[n]=(Z[1], …, Z[n])T denote the vectors of concomitants corresponding to X(n) ((Y[r], Z[r])T, for r=1, …, n, is the vector of bivariate concomitants corresponding to X(r)). We then present an alternate approach for the derivation of the exact joint distribution of (X0, X(r), Y[r], Z[r])T, for r=1, …, n. We show that these joint distributions can be expressed as mixtures of four-variate unified skew-elliptical distributions and these mixture forms facilitate the prediction of X(r), say, based on the concomitants Y[r] and Z[r]. Finally, we illustrate the usefulness of our results by a real data.
Acknowledgements
The authors express their sincere thanks to the associate editor and two anonymous reviewers for their valuable comments and suggestions on an earlier version of the manuscript which led to this improved version.