Minimax Nonparametric Two-sample Test under Smoothing
This work addresses the problem of nonparametric two-sample density testing for statisticians and data analysts, offering a minimax optimal method that improves upon existing approaches, though it is incremental in advancing smoothing spline frameworks.
The authors tackled the problem of comparing probability densities between two groups by developing a new probabilistic tensor product smoothing spline framework, which transforms density comparison into interaction testing. They proposed a penalized likelihood ratio test that is asymptotically chi-square distributed under the null, derived a sharp minimax testing rate, and showed their test is minimax optimal, with simulations and real applications demonstrating it outperforms conventional approaches.
We consider the problem of comparing probability densities between two groups. A new probabilistic tensor product smoothing spline framework is developed to model the joint density of two variables. Under such a framework, the probability density comparison is equivalent to testing the presence/absence of interactions. We propose a penalized likelihood ratio test for such interaction testing and show that the test statistic is asymptotically chi-square distributed under the null hypothesis. Furthermore, we derive a sharp minimax testing rate based on the Bernstein width for nonparametric two-sample tests and show that our proposed test statistics is minimax optimal. In addition, a data-adaptive tuning criterion is developed to choose the penalty parameter. Simulations and real applications demonstrate that the proposed test outperforms the conventional approaches under various scenarios.