Integral-Operator-Based Spectral Algorithms for Goodness-of-Fit Tests

TL;DR

This paper introduces a flexible spectral filtering framework for kernel-based goodness-of-fit tests, improving their ability to distinguish distributions with theoretical guarantees and competitive empirical performance.

Contribution

It generalizes existing discrepancy measures by removing restrictive assumptions, enabling more effective regularization for distribution testing.

Findings

Tests achieve valid Type I error control.

Enhanced power compared to existing methods.

Numerical experiments demonstrate broad applicability.

Abstract

The widespread adoption of the \emph{maximum mean discrepancy} (MMD) in goodness-of-fit testing has spurred extensive research on its statistical performance. However, recent studies indicate that the inherent structure of MMD may constrain its ability to distinguish between distributions, leaving room for improvement. Regularization techniques have the potential to overcome this limitation by refining the discrepancy measure. In this paper, we introduce a family of regularized kernel-based discrepancy measures constructed via spectral filtering. Our framework can be regarded as a natural generalization of prior studies, removing restrictive assumptions on both kernel functions and filter functions, thereby broadening the methodological scope and the theoretical inclusiveness. We establish non-asymptotic guarantees showing that the resulting tests achieve valid Type~I error control and enhanced power performance. Numerical experiments are conducted to demonstrate the broader generality and competitive performance of the proposed tests compared with existing methods.