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

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.