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In this article, we consider the problem of comparing several multivariate normal mean vectors when the covariance matrices are unknown and arbitrary positive definite matrices. We propose a parametric bootstrap (PB) approach, and develop an approximation to the distribution of the PB pivotal quantity for comparing two mean vectors. This approximate test is shown to be the same as the invariant test given in Krishnamoorthy and Yu (2004, Statistics & Probability Letters, 66, 161-169) for thedoi:10.1080/00949650902822564 fatcat:atrlqgreqbhsrkgxw3h2n4bddu