Dependence comparisons of order statistics in the proportional hazards model

Abstract

Let $X_1,\dots ,X_n$ be mutually independent exponential random variables with distinct hazard rates ${\lambda }_1,\dots ,{\lambda }_n>0$ and let $Y_1,\dots ,Y_n$ be a random sample from the exponential distribution with hazard rate $\bar \lmd = \sum_{i=1}^n \lmd_i/n$. Also let $X_{1:n}<\cdots <X_{n:n}$ and $Y_{1:n}<\cdots <Y_{n:n}$ be their associated order statistics. It is shown that for $1\le i<j\le n$, the generalized spacing $X_{j:n}-X_{i:n}$ is more dispersed than $Y_{j:n}-Y_{i:n}$ according to dispersive ordering. This result is used to solve a long standing open problem that for $2\le i\le n$ the dependence of $X_{i:n}$ on $X_{1:n}$ is less than that of $Y_{i:n}$ on $Y_{1:n}$, in the sense of the more stochastically increasing. This dependence result is also extended to the PHR model. This extends the earlier work of {\em Genest, Kochar and Xu}[ J.\ Multivariate Anal.\ {\bf 100} (2009) \ 1587-1592] who proved this result for $i=n$.