论文标题
比例危害模型中订单统计的依赖性比较
Dependence comparisons of order statistics in the proportional hazards model
论文作者
论文摘要
令$ x_1，\ ldots，x_n $是相互独立的指数随机变量，具有明显危险率$λ_1，\ ldots，λ_n> 0 $，让$ y_1，\ ldots，y_n $是从危险率$ \ bar \ bar \ lmd = sum______的$ y_n $的随机样本，是一个随机样本。另外，让$ x_ {1：n} <\ cdots <x_ {n：n} $和$ y_ {1：n} <\ cdots <y_ {n：n} $是他们的关联顺序统计信息。结果表明，对于$ 1 \ le i <j \ le n $，广义间距$ x_ {j：\，n} - x_ {i：\，n} $比$ y_ {j：\，n} - y_ y__ {y y_ {i：\，n} $比$ y_ {j：\，n} -n} $。该结果用于解决一个长期的开放问题，以$ 2 \ le i \ le n $，$ x_ {i：\，n} $的依赖性在$ x_ {1：\，n} $上小于$ y_ {i：\，n} $ of $ y_ {i：\，n} $ on $ y_ {1 \，：此依赖性结果也扩展到PHR模型。这扩展了{\ em genest，kochar和xu}的早期工作[J. \多变量肛门。
Let $X_1, \ldots , X_n$ be mutually independent exponential random variables with distinct hazard rates $λ_1, \ldots , λ_n > 0$ and let $Y_1, \ldots, Y_n$ 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 $i =n$.
