Prediction in a Trivariate Normal Distribution via Two Order Statistics

In this paper, assuming that (X, Y1, Y2)T has a trivariate normal distribution, we derive the exact joint distribution of ( X, Y(1), Y(2))^T, where Y(1) and Y(2) are order statistics arising from (Y1, Y2)T . We show that this joint distribution is a mixture of truncated trivariate normal distributions and then use this mixture representation to derive the best (nonlinear) predictiors of X in terms of ( Y(1), Y(2))^T. We also predict Y(1) in terms of ( X, Y(2) )^T , and Y(2) in terms of ( X, Y(1))^T. Finally illustrate the usefulness of these results by using real-life data.