A One-Sample Test for Normality with Kernel Methods

TL;DR

This paper introduces a kernel-based one-sample normality test in RKHS that is computationally efficient and effective in high-dimensional data, outperforming traditional methods.

Contribution

It presents a novel normality test using kernel methods and a specialized bootstrap, applicable to data and parameter testing, with improved performance in high dimensions.

Findings

Outperforms traditional normality tests in high-dimensional settings

Uses a computationally efficient bootstrap method

Provides bounds on Type-II error based on influential quantities

Abstract

We propose a new one-sample test for normality in a Reproducing Kernel Hilbert Space (RKHS). Namely, we test the null-hypothesis of belonging to a given family of Gaussian distributions. Hence our procedure may be applied either to test data for normality or to test parameters (mean and covariance) if data are assumed Gaussian. Our test is based on the same principle as the MMD (Maximum Mean Discrepancy) which is usually used for two-sample tests such as homogeneity or independence testing. Our method makes use of a special kind of parametric bootstrap (typical of goodness-of-fit tests) which is computationally more efficient than standard parametric bootstrap. Moreover, an upper bound for the Type-II error highlights the dependence on influential quantities. Experiments illustrate the practical improvement allowed by our test in high-dimensional settings where common normality tests are known to fail. We also consider an application to covariance rank selection through a sequential procedure.