The general outlier problem for a multivariate normal random sample with mean slippage is defined and is shown to be invariant under a natural group of transformations. A family of maximal invariants is obtained, and the common distribution of its members is derived. The critical region for the locally best invariant test of the null hypothesis, that there are no outliers, versus the alternative hypothesis, that some outliers are present, is found. Under very general conditions, this test is equivalent to rejecting the null hypothesis whenever Mardia's multivariate sample kurtosis is sufficiently large.
Steven J. Schwager. Barry H. Margolin. "Detection of Multivariate Normal Outliers." Ann. Statist. 10 (3) 943 - 954, September, 1982. https://doi.org/10.1214/aos/1176345884