Abstract: Abstract: We consider the problem of comparing probability densities among multiple groups. To this end, we develop a new probabilistic tensor product smoothing spline framework 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, for which we propose a penalized likelihood ratio test. Here we show that the test statistic is asymptotically chi-squared distributed under the null hypothesis. Furthermore, we derive a sharp minimax testing rate based on the Bernstein width for nonparametric multi-sample tests, and show that our proposed test statistic is minimax optimal. In addition, we develop a data-adaptive tuning criterion for choosing the penalty parameter. The results of simulations and real applications demonstrate that the proposed test outperforms conventional approaches under various scenarios.

Key words and phrases: Minimax optimality, multi-sample test, nonparametric test, penalized likelihood ratio test, smoothing splines, Wilks’ phenomenon.