Maximizing the probability of correctly ordering random variables using linear predictors

Let (T1, x1), (T2, x2), ..., (Tn, xn) be a sample from a multivariate normal distribution where Ti are (unobservable) random variables and xi are random vectors in Rk. If the sample is either independent and identically distributed or satisfies a multivariate components of variance model, then the probability of correctly ordering {Ti} is maximized by ranking according to the order of the best linear predictors {E(Tixi)}. Furthermore, it orderings are chosen according to linear functions {b'xi} then the conditional probability of correct order given (Ti = t1; I = 1, ..., n) is maximized when b'xi is the best linear predictor. Examples are given to show that linear predictors may not be optimal and that using a linear combination other that the best linear predictor may give a greater probability of correctly ordering {Ti} if {(Ti, xi)} are independent but not identically distributed, or if the distributions are not normal.