A unified framework for multivariate two-sample and k-sample kernel-based quadratic distance goodness-of-fit tests

TL;DR

This paper introduces a unified kernel-based framework for multivariate two-sample and k-sample goodness-of-fit tests, connecting quadratic distances with maximum mean discrepancy and providing practical algorithms with software implementations.

Contribution

It develops a comprehensive framework for multivariate goodness-of-fit testing using quadratic distances, extending to k-sample scenarios, with theoretical derivations and software tools.

Findings

The quadratic distance kernel test matches the maximum mean discrepancy form.

The k-sample tests have well-defined asymptotic distributions.

Simulations and real data demonstrate the tests' effectiveness.

Abstract

In the statistical literature, as well as in artificial intelligence and machine learning, measures of discrepancy between two probability distributions are largely used to develop measures of goodness-of-fit. We concentrate on quadratic distances, which depend on a non-negative definite kernel. We propose a unified framework for the study of two-sample and k-sample goodness of fit tests based on the concept of matrix distance. We provide a succinct review of the goodness of fit literature related to the use of distance measures, and specifically to quadratic distances. We show that the quadratic distance kernel-based two-sample test has the same functional form with the maximum mean discrepancy test. We develop tests for the $k$-sample scenario, where the two-sample problem is a special case. We derive their asymptotic distribution under the null hypothesis and discuss computational aspects of the test procedures. We assess their performance, in terms of level and power, via extensive simulations and a real data example. The proposed framework is implemented in the QuadratiK package, available in both R and Python environments.