# Minimax Nonparametric Two-sample Test under Smoothing

arXiv: https://arxiv.org/abs/1911.02171 · HTML: https://mlsanity.com/paper/1911.02171/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Nov 2019.

## Built on by (most cited first, 1 of 1)

- [Robust Flow-based Conformal Inference (FCI) with Statistical Guarantee](https://mlsanity.com/paper/2205.10732/) (2022)

## Abstract (arXiv)

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.