AI 中文总结
该研究针对独立非负随机变量，通过重加权概率测度刻画次序统计量与对应秩指标的独立性，得到相关显式表达式与独立性偏离度量，还在删失场景下完成相关刻画并验证了方法适用性。
AI 中文摘要
针对独立非负随机变量，我们研究次序统计量与其对应秩指标之间的独立性。通过一种概率测度建立了通用刻画，该测度是根据单个观测值占据指定秩的条件概率对其分布进行重加权得到的。该框架给出了相关加权函数、条件分布的显式表达式，以及基于Kolmogorov距离和Wasserstein距离的分布无关的独立性偏离度量。我们特别关注最小和最大次序统计量，在单删失和多删失机制下，得到了独立右删失和左删失的新刻画。我们还基于比例风险模型和比例反向风险模型建立了充分条件，并在经典单删失场景下推导了完整刻画。我们给出了离散和连续的例子以及数值说明，以证明所提方法的适用性。
英文摘要
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.
Comments20 pages, 2 figures
