Minimax Nonparametric Two-sample Test under Smoothing

TL;DR

This paper introduces a novel nonparametric two-sample test based on a tensor product smoothing spline framework, which effectively detects differences in joint densities with minimax optimality and adaptive tuning.

Contribution

It develops a new probabilistic tensor product smoothing spline framework and a penalized likelihood ratio test that is minimax optimal for nonparametric density comparison.

Findings

Test statistic is asymptotically chi-square distributed under null hypothesis.

Proposed test outperforms conventional methods in simulations and real data.

Achieves the sharp minimax testing rate based on Bernstein width.

Abstract

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.