全变差距离的高概率下界
High Probability Lower Bounds for the Total Variation Distance
摘要：
统计和机器学习社区最近对基于分类的双样本测试方法越来越感兴趣。基于分类的双样本检验的结果仍然是拒绝决策，这并不总是提供信息，因为原假设很少严格为真。因此，当测试拒绝时，提供额外的量作为分布差异的精确测量将是有益的。在这项工作中，我们介绍了一个框架的建设高概率的全变差距离的下界。这些界限基于一维投影，例如分类或回归方法，并且可以被解释为指向分布差异的样本的最小分数。我们进一步推导出渐近功率和检测率的两个建议的估计，并讨论潜在的用途，通过应用再分析气候数据集。
英文摘要：
The statistics and machine learning communities have recently seen a growing interest in classification-based approaches to two-sample testing. The outcome of a classification-based two-sample test remains a rejection decision, which is not always informative since the null hypothesis is seldom strictly true. Therefore, when a test rejects, it would be beneficial to provide an additional quantity serving as a refined measure of distributional difference. In this work, we introduce a framework for the construction of high-probability lower bounds on the total variation distance. These bounds are based on a one-dimensional projection, such as a classification or regression method, and can be interpreted as the minimal fraction of samples pointing towards a distributional difference. We further derive asymptotic power and detection rates of two proposed estimators and discuss potential uses through an application to a reanalysis climate dataset.
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