Dependence among order statistics for time-transformed exponential models

Abstract

Abstract Let $(X_{1},\ldots,X_{n})$ 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. Let for $1\leq i\leq n$ , $X_{i:n}$ denote the corresponding i th-order statistic. We consider the problem of comparing the strength of dependence between any pair of X i ’s with that of the corresponding order statistics. It is in particular proved that for $m=2,\ldots,n$ , the dependence of $X_{2:m}$ on $X_{1:m}$ is more than that of X 2 on X 1 according to more stochastic increasingness (positive monotone regression) order, which in turn implies that $(X_{1:m},X_{2:m})$ is more concordant than $(X_{1},X_{2})$ . It will be interesting to examine whether these results can be extended to other exchangeable models.