Resolving an open problem on the hazard rate ordering ofp-spacings

Abstract

AbstractLet$V_{(r,n,\tilde {m}_n,k)}^{(p)}$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 betweenp-spacings of generalized order statistics from two samples.Probability in the Engineering and Informational Sciences20: 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.