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Dependence comparisons of order statistics in the proportional hazards model
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- Journal:
- Probability in the Engineering and Informational Sciences / Volume 37 / Issue 3 / July 2023
- Published online by Cambridge University Press:
- 21 April 2022, pp. 730-736
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Let
$X_1, \ldots, X_n$ be mutually independent exponential random variables with distinct hazard rates 
$\lambda _1, \ldots, \lambda _n$ and let 
$Y_1, \ldots, Y_n$ be a random sample from the exponential distribution with hazard rate 
$\bar \lambda = \sum _{i=1}^{n} \lambda _i/n$. Also let 
$X_{1:n} \lt \cdots \lt X_{n:n}$ and 
$Y_{1:n} \lt \cdots \lt Y_{n:n}$ be their associated order statistics. It is proved that for 
$1\le i \lt j \le n$, the generalized spacing 
$X_{j:n} - X_{i:n}$ is more dispersed than 
$Y_{j:n} - Y_{i:n}$ according to dispersive ordering and 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 ordering. This dependence result is also extended to the proportional hazard rates (PHR) model. This extends the earlier work of Genest et al. [(2009)]. On the range of heterogeneous samples. Journal of Multivariate Analysis 100: 1587–1592] who proved this result for 
$i =n$.
