On the Rates of Convergence of Induced Ordered Statistics and their Applications
诱导次序统计量的收敛速率及其应用AI 解读 ↓
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诱导次序统计量的收敛速率及其应用
- 研究问题
- 按辅助变量排序后，邻近观测的响应分布能以何种速率逼近目标条件分布，且允许使用的近邻数量如何随样本规模增长？
- 方法与思路
- 作者在二次均值可微条件下推导 Hellinger 与全变差距离的边际速率，再建立联合收敛结果，并补充另一类余项条件下的分析。
- 材料陈述的贡献
- 摘要称结果同时容许内部与边界条件点，揭示平滑度和收敛速度的权衡、两种距离的差异，并给出近邻数量的增长条件。
- 阅读关注点
- 应核对边界点的可微条件、边际误差向联合误差转化的步骤，以及近邻增长限制在回归不连续和其他应用中的具体含义。
- 阅读边界
- 这是 arXiv 预印本，不能据此认定已同行评审或被期刊接收；摘要未给出速率公式与完整条件，所称较弱假设仍须结合具体分布核验。
原文摘要
Induced order statistics (IOS) arise when sample units are reordered according to the value of an auxiliary variable, and the associated responses are analyzed in that induced order. IOS play a central role in applications where the goal is to approximate the conditional distribution of an outcome at a fixed covariate value using observations whose covariates lie closest to that point, including regression discontinuity designs, k-nearest-neighbor methods, and distributionally robust optimization. Existing asymptotic results allow the dimension of the IOS vector to grow with the sample size only under smoothness conditions that are often too restrictive for practical data-generating processes. In particular, these conditions rule out boundary points, which are central to regression discontinuity designs. This paper develops general convergence rates for IOS under primitive and comparatively weak assumptions. We derive sharp marginal rates for the approximation of the target conditional distribution in Hellinger and total variation distances under quadratic mean differentiability and show how these marginal rates translate into joint convergence rates for the IOS vector. Our results are widely applicable: they rely on a standard smoothness condition and accommodate both interior and boundary conditioning points, as required in regression discontinuity and related settings. In the supplementary appendix, we provide complementary results under a Taylor/Holder remainder condition. Our results reveal a clear trade-off between smoothness and speed of convergence, identify regimes in which Hellinger and total variation distances behave differently, and provide explicit growth conditions on the number of nearest neighbors.