Let $X_1, \ldots , X_n$ be mutually independent exponential random variables with distinct hazard rates $\lambda_1, \ldots , \lambda_n > 0$ 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 $i =n$.
翻译:$X_ 1,\ ldots, X_n$ 是相互独立的指数性随机变量, 其危害率不同 $\ lambda_ 1,\ldots,\lambda_ n > 0美元, 并且让Y_ 1,\ldts, Y_n$是指数分布的随机样本, 危险率为$\bar\ lmd=\ sum_ i= 1\\\\\ lmd_ i/ n。 另外, $X% 1:n} < cdots < Xn: n} 美元和$Y1:n} < coldots > < Y@n: n} 美元是相关的顺序统计。 显示对于 $ < i\\\ 美元=leg\ n美元, 通用的间隔 $X\\\\\\\\\\\\\\\\\\\ n\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\