#### Published In

Probability in Engineering and Information Sciences

#### Document Type

Pre-Print

#### Publication Date

3-17-2022

#### Subjects

Probability and statistics

#### Abstract

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 $\bar \lmd = \sum_{i=1}^n \lmd_i/n$. 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 {\em Genest, Kochar and Xu}[ J.\ Multivariate Anal.\ {\bf 100} (2009) \ 1587-1592] who proved this result for *i*=*n*.

#### Rights

© Cambridge University Press 2022

#### Persistent Identifier

https://archives.pdx.edu/ds/psu/37228

#### Citation Details

Kochar, Subhash C., (2022). [PRE-PRINT] "Dependence Comparisons of Order Statistics in the Proportional Hazards Model," To be published in *Probability in Engineering and Information Sciences*.

## Description

This is a pre-copyedited, author-produced PDF of an article accepted for publication in

Probability in Engineering and Information Sciencesfollowing peer review. The version of record will be available online at: https://www.cambridge.org/core/journals/probability-in-the-engineering-and-informational-sciences