Multivariate normality test based on the uniform distribution on the Stiefel manifold

TL;DR

This paper introduces a novel statistical test for multivariate normality utilizing the uniform distribution on the Stiefel manifold, with exact distribution properties and simulation-based validation.

Contribution

It proposes a new test statistic based on the Stiefel manifold, providing exact distribution under the null hypothesis for multivariate normality testing.

Findings

Test statistic follows a matrix-variate normal distribution under null hypothesis

Monte Carlo simulations confirm accurate Type I error control

Demonstrates good power in non-asymptotic scenarios

Abstract

This study presents a new procedure for necessary tests of multivariate normality based on the uniform distribution on the Stiefel manifold. We demonstrate that the test statistic, which is formed by the product of the scaled residual matrix and the symmetric square root of a Wishart matrix, is exactly distributed as a matrix-variate normal distribution under the null hypothesis. Monte Carlo simulations are conducted to assess the Type I error rate and power in non-asymptotic settings.