TITLE:
Bootstrap Implementation of Mardia’s K² Test and Performance Comparison with Royston’s H Test
AUTHORS:
José Moral de la Rubia
KEYWORDS:
Multivariate Normal Distribution, Statistical Hypothesis Testing, Statistical Power, Multivariate Skewness, Multivariate Kurtosis
JOURNAL NAME:
Journal of Data Analysis and Information Processing,
Vol.14 No.1,
February
5,
2026
ABSTRACT: This article has two objectives. The first is to develop an R script that performs Mardia’s K2 test for assessing multivariate normality (MVN), using both its original asymptotic formulation and a bootstrap method. Unlike previous studies that rely on Monte Carlo simulations to obtain critical values, the bootstrap approach focuses on resampling from sample data while preserving its correlational structure. The second objective is to compare both variants of K2 test with Royston’s H test in terms of hit rate and statistical power. A total of 200 samples were generated from MVN distributions and another 200 from multivariate t distributions with five degrees of freedom, varying sample size (7 levels), number of variables (5 levels), and homogeneous correlation among variables (5 levels). The script computes the K2 and H test statistics, their p-values, and statistical power, and also provides graphical representations of the sampling distribution of the K2 statistic. Two worked examples, one based on a randomly generated sample from an MVN distribution and another based on an empirical sample without an MVN distribution, demonstrate the script’s functionality. The sampling distribution of the bootstrap K2 statistic is less peaked and exhibits a heavier right tail than the distribution implied by the asymptotic approximation, which is defined by a weighted sum of correlated chi-squared variables. When the null hypothesis was retained (i.e., under multivariate normality), the H test showed slightly better performance in terms of correct retention. In contrast, when the null hypothesis was violated, both variants of Mardia’s K2 test outperformed the H test; in this context, the asymptotic variant generally performed better than the bootstrap variant. In samples drawn from a multivariate t distribution, as well as in analyses aggregated across both distributions, the asymptotic K2 variant achieved the highest mean power. In multivariate normal samples, however, the bootstrap K2 variant showed superior performance, followed by the asymptotic variant, with the H test performing worst. This new variant is a robust option for assessing MVN, and its use and further exploration are recommended.