When an n × 1 random vector X = (X1, …, Xn ) T has a sign-invariant distribution, Strait [J. Multivariate Anal. 4 (1974) 494-496] proved that the expectations of max(0, X1, X1 + X2, …, X1 + Xn ) and max(0, X1, …, Xn ) are equal. In this note we assume a weaker condition that (X1, X2, …, Xn ) and (-X1, X2, …, Xn ) are equal in distribution and prove a more general result that the expectations of Lr (0, X1, X1 + X2, …, X1 + Xn ) and Lr (0, X1, …, Xn ) are equal, where Lr (0, X1, …, Xn ) is the rth order statistic of 0, X1, …, Xn for r = 1, …, n + 1.

中文翻译：

---

一组随机变量的顺序统计的恒等式

当 n × 1 随机向量 X = (X1, …, Xn ) T 具有符号不变分布时，Strait [J. 多元肛门。4 (1974) 494-496] 证明了 max(0, X1, X1 + X2, …, X1 + Xn ) 和 max(0, X1, …, Xn ) 的期望是相等的。在本笔记中，我们假设 (X1, X2, …, Xn ) 和 (-X1, X2, …, Xn ) 在分布上相等，并证明了一个更一般的结果，即 Lr (0, X1, X1 + X2, ..., X1 + Xn ) 和 Lr (0, X1, ..., Xn ) 相等，其中 Lr (0, X1, ..., Xn ) 是 r = 0, X1, ..., Xn 的 r 阶统计量1, …, n + 1。