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Probability in Engineering and Information Sciences

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Probability and statistics


Let X1,…,Xn be mutually independent exponential random variables with distinct hazard rates λ1,…,λn > 0 and let Y1,…,Yn be a random sample from the exponential distribution with hazard rate $\bar \lmd = \sum_{i=1}^n \lmd_i/n$. Also let X1:n < ⋯ < Xn:n and Y1:n < ⋯ < Yn:n be their associated order statistics. It is shown that for 1 i < j n, the generalized spacing Xj:n - X i:n is more dispersed than Yj:n− Yi:n according to dispersive ordering. This result is used to solve a long standing open problem that for 2 ≤ i n the dependence of Xi:n on X1:n is less than that of Yi:n on Y1: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.


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