Regression analysis using order statistics

Abstract

In this article, we study the joint distribution of X and two linear combinations of order statistics, aTY(2) and bTY(2), where a = (a1, a2)T and b = (b1, b2)T are arbitrary vectors in R2 and Y(2) = (Y(1), Y(2))T is a vector of ordered statistics obtained from (Y1, Y2)T when (X, Y1, Y2)T follows a trivariate normal distribution with a positive definite covariance matrix. We show that this distribution belongs to the skew-normal family and hence our work is a generalization of Olkin and Viana (J Am Stat Assoc 90:1373–1379, 1995) and Loperfido (Test 17:370–380, 2008).