Dependence comparisons of order statistics in the proportional hazards model

Let $X_1, \ldots, X_n$ be mutually independent exponential random variables with distinct hazard rates $\lambda _1, \lambda _n$ and let $Y_1, Y_n$ a sample from the distribution rate $\bar = \sum _{i=1}^{n} _i/n$ . Also $X_{1:n} \lt \cdots X_{n:n}$ $Y_{1:n} Y_{n:n}$ their associated order statistics. It is proved that for $1\le i j \le n$ , generalized spacing $X_{j:n} - X_{i:n}$ more dispersed than $Y_{j:n} Y_{i:n}$ according to dispersive ordering $2\le dependence of $X_{i:n}$ on $X_{1:n}$ less $Y_{i:n}$ $Y_{1 :n}$ in sense stochastically increasing ordering. This result also extended proportional (PHR) model. extends earlier work Genest et al. [(2009)]. On range heterogeneous samples. Journal Multivariate Analysis 100: 1587–1592] who this $i =n$