Outlier test for a group of multivariate observations

Assume that we have m independent random samples each of size n from Np(; ) and our goal is to test whether or not the ith sample is an outlier (i=1,2,…..m). To date it is well known that a test statistics exist whose null distribution is Betta and given the relationship between Betta and F distribution, an F test statistic can be used. In the statistical literature however a clear and precise proof is not accessible and in some cases the proof is incomplete. In this paper a precise and relatively clear proof is given and through simulation, capability and weakness of the test is considered.