To impute or not? Testing multivariate normality on incomplete dataset: revisiting the BHEP test

In this paper, we focus on testing multivariate normality using the BHEP test with data that are missing completely at random. Our objective is twofold: first, to gain insight into the asymptotic behavior of the BHEP test statistics under two widely used approaches for handling missing data, namely complete-case analysis and imputation, and second, to compare the power performance of the test statistic under these approaches. Since complete-case approach removes all elements of the sample with at least one missing component, it might lead to the loss of information. On the other hand, we note that performing the test on imputed data as if they were complete, Type I error becomes severely distorted. To address these issues, we propose an appropriate bootstrap algorithm for approximating p-values. Extensive simulation studies demonstrate that both mean and median approaches exhibit greater power compared to testing with complete-case analysis, and open some questions for further research. The proposed methodology is illustrated with real-data examples.