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

Published online by Cambridge University Press:  11 November 2022

Mahdi Alimohammadi Open the ORCID record for Mahdi Alimohammadi [Opens in a new window]
Mahdi Alimohammadi*
Affiliation:
Department of Statistics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran. E-mail: m.alimohammadi@alzahra.ac.ir
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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.

Keywords

LogconvexityStochastic comparisonsTotal positivity
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 37 , Issue 4 , October 2023 , pp. 1020 - 1028
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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