On the Convex Transform Order of Order Statistics
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.
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BibTeX
@misc{li2026convex,
title = {{On the Convex Transform Order of Order Statistics}},
author = {Yijia Li and Maochao Xu and Peng Zhao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26595},
eprint = {2609.26595},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26595},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - On the Convex Transform Order of Order Statistics
AU - Yijia Li
AU - Maochao Xu
AU - Peng Zhao
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26595
UR - https://arxiv.org/abs/2609.26595
AB - 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.
ER -