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

Koki Shimizu, Toshiya Iwashita

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.

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

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.
Paper Structure (4 sections, 3 theorems, 15 equations, 3 tables, 1 algorithm)

This paper contains 4 sections, 3 theorems, 15 equations, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Uniform Distribution on the Stiefel Manifold and its properties
  3. Test statistic for multivariate normality and testing procedure
  4. Simulations

Key Result

Lemma 1

Let $Z_0$ be an $n \times p$ random matrix. Its unique polar decomposition of $Z_0$ is given by Then, the random matrix ${Z_0}$ is distributed as matrix-variate normal distribution $N(O, I_n\otimes I_p)$ (see, chikuse2003statistics, p. 23 for details) if and only if the following conditions are satisfied:

Theorems & Definitions (6)

  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Remark 1