Dependence comparisons of order statistics in the proportional hazards model

TL;DR

This paper compares the dependence structures of order statistics in exponential models, showing that the proportional hazards model exhibits less dependence than the standard exponential model, with implications for statistical dependence measures.

Contribution

It establishes a new dependence comparison between order statistics in the proportional hazards model and the exponential model, extending previous results to a broader class.

Findings

Generalized spacings are more dispersed in the exponential model.

Dependence of order statistics on the minimum is weaker in the proportional hazards model.

Results extend to the proportional hazards regression model.

Abstract

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 $i =n$.