Dependence comparisons of order statistics in the proportional hazards model

Abstract

Let $X_1, \ldots, X_n$ be mutually independent exponential random variables with distinct hazard rates $\lambda _1, \ldots, \lambda _n$ and let $Y_1, \ldots, Y_n$ be a random sample from the exponential distribution with hazard rate $\bar \lambda = \sum _{i=1}^{n} \lambda _i/n$ . Also let $X_{1:n} < \cdots < X_{n:n}$ and $Y_{1:n} < \cdots < Y_{n:n}$ be their associated order statistics. It is proved 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 and 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 ordering. This dependence result is also extended to the proportional hazard rates (PHR) model. This extends the earlier work of Genest et al. [(2009)]. On the range of heterogeneous samples. Journal of Multivariate Analysis 100: 1587–1592] who proved this result for $i =n$ .