Parametric test based on the bootstrapping approach for the MANOVA under a Behrens-Fisher problem

This article presents a comparison of multivariate normal mean vectors under covariance positive definite matrices. We introduce an improved parametric bootstrap (IPB) approach for addressing the multivariate Behrens-Fisher problem, specifically focusing on cases with unequal covariance matrices. Additionally, we evaluate the performance of the IPB test by comparing it with three existing tests: the parametric bootstrap (PB) test, the generalized variable (GV) test, and the Johansen test. Through Monte Carlo simulation, our results demonstrate that both the IPB test and the PB test exhibit superior control over Type I error rates compared to the GV and Johansen tests. Notably, the IPB test outperforms the PB test in terms of controlling Type I error rates. Consequently, our study concludes that the IPB test represents a preferred statistical method for testing the equality of mean vectors in the multivariate Behrens-Fisher problem.