Asymptotics and practical aspects of testing normality with kernel methods

TL;DR

This paper analyzes the asymptotic behavior and practical implementation of a kernel-based normality test in Hilbert spaces, emphasizing its consistency, null distribution approximation, and performance in high-dimensional, low-sample scenarios.

Contribution

It provides the first detailed asymptotic analysis of the kernel-based normality test and offers a concrete method for approximating its null distribution in practice.

Findings

The test is asymptotically normal under fixed alternatives.

The null distribution can be reliably approximated using a derived integral kernel.

The method performs well in high-dimensional, low-sample size settings.

Abstract

This paper is concerned with testing normality in a Hilbert space based on the maximum mean discrepancy. Specifically, we discuss the behavior of the test from two standpoints: asymptotics and practical aspects. Asymptotic normality of the test under a fixed alternative hypothesis is developed, which implies that the test has consistency. Asymptotic distribution of the test under a sequence of local alternatives is also derived, from which asymptotic null distribution of the test is obtained. A concrete expression for the integral kernel associated with the null distribution is derived under the use of the Gaussian kernel, allowing the implementation of a reliable approximation of the null distribution. Simulations and applications to real data sets are reported with emphasis on high-dimension low-sample size cases.