Resolving an open problem on the hazard rate ordering of p-spacings

Abstract

Let $V_{(r,n,\tilde {m}_n,k)}^{(p)}$ and $W_{(r,n,\tilde {m}_n,k)}^{(p)}$ be the $p$ -spacings of generalized order statistics based on absolutely continuous distribution functions $F$ and $G$ , respectively. Imposing some conditions on $F$ and $G$ and assuming that $m_1=\cdots =m_{n-1}$ , Hu and Zhuang (2006. Stochastic orderings between p-spacings of generalized order statistics from two samples. Probability in the Engineering and Informational Sciences 20: 475) established $V_{(r,n,\tilde {m}_n,k)}^{(p)} \leq _{{\rm hr}} W_{(r,n,\tilde {m}_n,k)}^{(p)}$ for $p=1$ and left the case $p\geq 2$ as an open problem. In this article, we not only resolve it but also give the result for unequal $m_i$ 's. It is worth mentioning that this problem has not been proved even for ordinary order statistics so far.

Some New Results on Stochastic Comparisons of Spacings of Generalized Order Statistics from One and Two Samples

“…These notions are applied in many branches of statistical theory, with special attention to reliability and life testing. Stochastic comparisons of GOSs and their spacings have been discussed by several scholars (see, for instance, the investigations by Franco et al [3], Belzunce et al [4], Hu and Zhuang [5,6], Zhao and Balakrishnan [7], Balakrishnan et al [8] and Alimohammadi [9]).…”