Dependence Among Order Statistics for Time-transformed Exponential Models

Let (X1, . . . ,Xn) be a random vector distributed according to a time-transformed exponential model. This is a special class of exchangeable models, which, in particular, includes multivariate distributions with Schur-constant survival functions and with identical marginals. Let for 1 ≤ i ≤ n, Xi:n denote the corresponding ith order statistic. We consider the problem of comparing the strength of dependence between any pair of Xi’s with that of the corresponding order statistics. It is proved that for m = 2, . . . , n, the dependence of X2:m on X1:m is more than that of X2 on X1 according to more stochastic increasingness (positive monotone regression) order, which in turn implies that (X1:m,X2:m) is more concordant than (X1,X2). It will be interesting to examine whether these results can be extended to other exchangeable models