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A power study of goodness-of-fit tests for multivariate normality implemented in R

Multivariate statistical analysis procedures often require data to be multivariate normally distributed. Many tests have been developed to verify if a sample could indeed have come from a normally distributed population. These tests do not all share the same sensitivity for detecting departures from normality, and thus a choice of test is of central importance. This study investigates through simulated data the power of those tests for multivariate normality implemented in the statistic software R and pits them against the variant of testing each marginal distribution for normality. The results of testing two-dimensional data at a level of significance α=5% showed that almost one-third of those tests implemented in R do not have a type I error below this. Other tests outperformed the naive variant in terms of power even when the marginals were not normally distributed. Even though no test was consistently better than all alternatives with every alternative distribution, the energy-statistic test always showed relatively good power across all tested sample sizes.