Dependence comparisons of order statistics in the proportional hazards model

02/16/2022

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by   Subhash Kochar, et al.

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Let X_1, … , X_n be mutually independent exponential random variables with distinct hazard rates λ_1, … , λ_n > 0 and let Y_1, …, Y_n be a random sample from the exponential distribution with hazard rate = ∑_i=1^n _i/n. Also let X_1:n < ⋯ < X_n:n and Y_1:n < ⋯ < Y_n:n be their associated order statistics. It is shown that for 1≤ i <j ≤ 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≤ i ≤ 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 Genest, Kochar and Xu[ J. Multivariate Anal. 100 (2009) 1587-1592] who proved this result for i =n.