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 $X_{1:n} \lt \cdots \lt X_{n:n}$ and $Y_{1:n} \lt \cdots \lt Y_{n:n}$ be their associated order statistics. It is proved that for $1\le i \lt j \le n$, 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 $i =n$.

令$X_1, \ldots, X_n$为相互独立的指数随机变量，具有不同的风险率$\lambda _1, \ldots, \lambda _n$并令$Y_1, \ldots, Y_n$为指数分布的随机样本危险率$\bar \lambda = \sum _{i=1}^{n} \lambda _i/n$。还令$X_{1:n} \lt \cdots \lt X_{n:n}$和$Y_{1:n} \lt \cdots \lt Y_{n:n}$为其关联的订单统计数据。证明对于$1\le i \lt j \le n$，广义间距$X_{j:n} - X_{i:n}$比$Y_{j:n} - Y_{i更分散:n}$根据分散排序并且对于$2\le i \le n$， $X_{i:n}$对$X_{1:n}$的依赖性小于$Y_{i:n}$对$Y_{1 :n}$ 的依赖性，即更随机增加排序。这种依赖性结果也扩展到比例危险率（PHR）模型。这扩展了 Genest等人的早期工作。[（2009）]。关于异质样本的范围。Journal of Multivariate Analysis 100: 1587–1592] 谁证明了$i =n$的这个结果。