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.

设$V_{(r,n,\tilde {m}_n,k)}^{(p)}$和$W_{(r,n,\tilde {m}_n,k)}^{(p) }$分别是基于绝对连续分布函数$F$和$G$的广义阶次统计的$p$间距。对$F$和$G$施加一些条件，并假设$m_1=\cdots =m_{n-1}$，Hu 和 Zhuang (2006。来自两个样本的广义阶统计量的p间距之间的随机排序。概率工程与信息科学20: 475) 建立了$V_{(r,n,\tilde {m}_n,k)}^{(p)} \leq _{{\rm hr}} W_{(r,n) ,\tilde {m}_n,k)}^{(p)}$对于$p=1$并将$p\geq 2$的情况留为未决问题。在本文中，我们不仅解决了这个问题，还给出了不相等的$m_i$的结果。值得一提的是，迄今为止，即使对于普通的订单统计，这个问题也没有得到证明。