Asian Journal of Probability and Statistics

Testing for multivariate normality is a fundamental step in multivariate statistical analysis, as many classical techniques rely on this assumption. However, the arbitrary use of the multivariate normality test often leads to type I or type II error, which necessitates the review of the techniques for classification as Uniformly Most Powerful (UMP) and Locally Most Powerful (LMP). This study employed Monte Carlo simulations to investigate the empirical type I error rates and rejection powers of nine multivariate normality tests, including Shapiro–Wilk (MVSW), Energy test, Mardia’s test, Henze–Zirkler test, Zhang’s test, Robust Mahalanobis Distance, Royston’s H test, Doornik–Hansen test, and the High-Dimensional Energy test. Simulations were conducted using 1,000 replications at varying sample sizes (n = 15, 20, 25, 50, 100, 200) with dimension d = 3, on a multivariate normal distribution (MVN) and a multivariate t-distribution (MVT). The results showed that while all tests approached nominal error rates at large sample sizes, some, particularly the Energy test, Zhang’s test, Henze–Zirkler test, and the High-Dimensional Energy test, exhibited higher sensitivity to heavy-tailed alternatives, maintaining strong power across all sample sizes. These were classified as Uniformly Most Powerful (UMP). In contrast, tests such as Shapiro–Wilk, Royston’s H, and Robust Mahalanobis Distance were more conservative in small samples but effective in larger ones, thus categorised as Locally Most Powerful (LMP). The findings provide a practical classification framework that enables researchers to select appropriate tests based on sample size, data characteristics, and desired power properties.