Characterizations of independence between order statistics and rank indicators

Roberto Vila; Frederico Almeida

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arXiv · 2608.05428

Characterizations of independence between order statistics and rank indicators

Roberto Vila·Frederico Almeida

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Abstract

We study the independence between an order statistic and its corresponding rank indicator for independent nonnegative random variables. A general characterization is established through a probability measure obtained by reweighting the distribution of a single observation according to its conditional probability of occupying a prescribed rank. This framework yields explicit expressions for the associated weighting functions and conditional distributions, as well as distribution-free measures of departure from independence based on Kolmogorov and Wasserstein distances. Special attention is devoted to the minimum and maximum order statistics, leading to new characterizations of independent right- and left-censoring under both single and multiple censoring mechanisms. We also establish sufficient conditions based on proportional hazards and proportional reversed hazards models and derive complete characterizations in the classical single-censoring setting. Discrete and continuous examples, together with numerical illustrations, are presented to demonstrate the applicability of the proposed methodology.

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Roberto Vila, Frederico Almeida. 2026-08-05. Characterizations of independence between order statistics and rank indicators. https://arxiv.org/abs/2608.05428

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