On the Optimality of Gaussian Kernel Based Nonparametric Tests against Smooth Alternatives

TL;DR

This paper analyzes the asymptotic properties of Gaussian kernel-based nonparametric tests, demonstrating their minimax optimality against smooth alternatives and providing practical guidelines for choosing the kernel scaling parameter.

Contribution

It establishes the theoretical optimality of Gaussian kernel tests for various goodness-of-fit, homogeneity, and independence testing scenarios, with data-driven parameter selection strategies.

Findings

Gaussian kernel tests are minimax optimal against smooth alternatives.

Choosing a diverging scaling parameter is crucial for test performance.

A data-driven method for selecting the kernel scale improves practical test effectiveness.

Abstract

Nonparametric tests via kernel embedding of distributions have witnessed a great deal of practical successes in recent years. However, statistical properties of these tests are largely unknown beyond consistency against a fixed alternative. To fill in this void, we study here the asymptotic properties of goodness-of-fit, homogeneity and independence tests using Gaussian kernels, arguably the most popular and successful among such tests. Our results provide theoretical justifications for this common practice by showing that tests using Gaussian kernel with an appropriately chosen scaling parameter are minimax optimal against smooth alternatives in all three settings. In addition, our analysis also pinpoints the importance of choosing a diverging scaling parameter when using Gaussian kernels and suggests a data-driven choice of the scaling parameter that yields tests optimal, up to an iterated logarithmic factor, over a wide range of smooth alternatives. Numerical experiments are also presented to further demonstrate the practical merits of the methodology.