In this paper Ferguson's univariate normal results for detection of outliers with variance slippage are extended to the multivariate elliptically symmetric case with dispersion slippage. The locally optimum test we derive possesses all three robustness properties, optimality, null and nonnull, and is based on Mardia's multivariate kurtosis statistic.
"Detection of Multivariate Outliers with Dispersion Slippage in Elliptically Symmetric Distributions." Ann. Statist. 14 (4) 1619 - 1624, December, 1986. https://doi.org/10.1214/aos/1176350183