On a possible improvement of the D’Agostino test of normality

Using optimal transport theory, we propose a bivariate Monte Carlo critical region for transformed skewness and kurtosis underlying the D’Agostino test of normality. The p-value of the improved D’Agostino test corresponds to a naturally defined nonnormality score admitting a decomposition in terms of skewness and kurtosis deviations from normality. In a simulation study, we show that the improved D’Agostino test has almost the same power as the celebrated Shapiro-Wilk test in all simulation setups and that it clearly outperforms the Shapiro-Wilk test against a platykurtic alternative. We conclude by observing that the proposed method, with a multivariate “Monte Carlo mosaic” critical region, can be used as a general goodness-of-fit test also for other uni- and multivariate distributions with only minimal and straightforward modifications.