On the Convex Transform Order of Order Statistics

2026-09-22

AI Summary

研究通过凸变换顺序比较了异质和同质指数随机变量的顺序统计量，探讨了异质性如何改变顺序统计量的分布形状。

Abstract

Let X1,...,Xn be independent exponential random variables with arbitrary positive, not necessarily equal, rates, and let Y1,\ldots,Yn be independent and identically distributed exponential random variables. We prove that, for every order statistic, the homogeneous order statistic is smaller than its heterogeneous counterpart in the convex transform order. Our results show that, relative to the homogeneous benchmark, heterogeneity stretches upper quantiles more strongly than lower quantiles, while the homogeneous order statistic ages faster in the convex-transform sense. We further extend the comparison to a proportional-hazards family that includes Weibull, Lomax, and Burr XII distributions. We give explicit conditions under which the comparison is preserved and show that the shape restriction is sharp within the Weibull family. These results characterize how component heterogeneity changes the distributional shape of order statistics beyond effects on location or scale.

Problem

• Convex Transform Order
• Order Statistics
• Heterogeneity
• Exponential Random Variables
• Proportional-Hazards Family

Innovation

• convex transform order
• order statistics
• heterogeneity
• proportional-hazards family
• distributional shape