[论文解读] Efficiency Lower Bounds for Distribution-Free Hotelling-Type Two-Sample Tests Based on Optimal Transport

Nabarun Deb, Bhaswar B. Bhattacharya+1 more|arXiv (Cornell University)|Apr 5, 2021

Statistical Methods and Inference参考文献 117被引用 16

一句话总结

本文提出基于最优传输的多变量、非参数、完全分布自由检验，其渐近效率下限在Konijn替代假设下与Hotelling的$T^2$和Wilks似然比检验相当。该方法将Hodges-Lehmann与Chernoff-Savage效率结果扩展至多变量设置，且无需假设参数模型或对潜在分布的先验知识。

ABSTRACT

The Wilcoxon rank-sum test is one of the most popular distribution-free procedures for testing the equality of two univariate probability distributions. One of the main reasons for its popularity can be attributed to the remarkable result of Hodges and Lehmann (1956), which shows that the asymptotic relative efficiency of Wilcoxon's test with respect to Student's $t$-test, under location alternatives, never falls below 0.864, despite the former being exactly distribution-free for all sample sizes. Even more striking is the result of Chernoff and Savage (1958), which shows that the efficiency of a Gaussian score transformed Wilcoxon's test, against the $t$-test, is lower bounded by 1. In this paper we study the two-sample problem in the multivariate setting and propose distribution-free analogues of the Hotelling $T^2$ test (the natural multidimensional counterpart of Student's $t$-test) based on optimal transport and obtain extensions of the above celebrated results over various natural families of multivariate distributions. Our proposed tests are consistent against a general class of alternatives and satisfy Hodges-Lehmann and Chernoff-Savage-type efficiency lower bounds, despite being entirely agnostic to the underlying data generating mechanism. In particular, a collection of our proposed tests suffer from no loss in asymptotic efficiency, when compared to Hotelling $T^2$. To the best of our knowledge, these are the first collection of multivariate, nonparametric, exactly distribution-free tests that provably achieve such attractive efficiency lower bounds. We also demonstrate the broader scope of our methods in optimal transport based nonparametric inference by constructing exactly distribution-free multivariate tests for mutual independence, which suffer from no loss in asymptotic efficiency against the classical Wilks' likelihood ratio test, under Konijn alternatives.

研究动机与目标

• 开发适用于多变量、非参数、完全分布自由的两样本位置问题检验方法，且在不假设参数模型的前提下保持高渐近效率。
• 利用最优传输将Hodges-Lehmann与Chernoff-Savage效率下限——此前仅在单变量设置中成立——扩展至多变量情形。
• 构建对一般替代假设一致的检验方法，同时对潜在数据生成机制保持无偏。
• 证明这些检验在渐近效率方面不逊于经典参数检验（如Hotelling的$T^2$和Wilks似然比检验）。
• 通过实现多变量数据中相互独立性的分布自由检验，拓展最优传输在非参数推断中的应用范围。

提出的方法

• 利用最优传输映射定义对分布自由变换不变的多变量秩统计量。
• 基于经验测度之间的Wasserstein距离构造检验统计量，确保在原假设下具有精确的分布自由性质。
• 对基于传输的秩应用高斯得分变换，以实现效率提升，其原理类似于单变量情形下的Wilcoxon检验。
• 推导在原假设与局部替代假设下的渐近分布，以建立一致性和效率下限。
• 采用Konijn替代假设的概念，评估多变量独立性检验框架中的渐近相对效率。
• 运用经验过程理论与最优传输的工具，证明有限样本与渐近的分布自由性质。

实验结果

研究问题

• RQ1 能否构造出多变量、非参数、完全分布自由的两样本检验，使其渐近效率与Hotelling的$T^2$相当？
• RQ2 基于最优传输的秩统计量是否在多变量设置中保持Hodges-Lehmann与Chernoff-Savage效率下限？
• RQ3 此类检验能否在不损失效率的前提下扩展至多变量数据中相互独立性的检验？
• RQ4 在局部替代假设下，与经典参数检验相比，基于传输的检验的渐近相对效率如何？
• RQ5 所提出的框架能否在不假设参数形式的前提下，实现精确的分布自由推断，同时在一般多变量分布族中保持高统计功效？

主要发现

• 所提出的基于最优传输的两样本检验为完全分布自由，且在位置平移假设下，其渐近相对效率至少为1，与Hotelling的$T^2$相当。
• 对于一类多变量分布，检验维持了Chernoff-Savage效率下限1，其效率与单变量情形下经高斯得分变换的Wilcoxon检验相当。
• 检验对包括复杂依赖结构在内的广义替代假设具有一致性。
• 该方法的一个变体在Konijn替代假设下，与Wilks似然比检验相比，在渐近效率方面无任何损失。
• 这是首个能严格证明实现此类强效率下限的多变量、非参数、分布自由检验。
• 该框架表明，最优传输可实现非参数多变量推断中的有限样本精确性与高效率，且无需参数假设。