Bounds for the largest Mahalanobis distance

Upper and lower bounds for the magnitude of the largest Mahalanobis distance, calculated from n multivariate observations of length p, are derived. These bounds are multivariate extensions of corresponding bounds that arise for the most deviant Z-score calculated from a univariate sample of size n. The approach taken is to pose optimization problems in a mathematical context and to employ variational methods to obtain solutions. The attainability of the bounds obtained is demonstrated. Bounds for related quantities (elements of the “hat matrix”) are also derived.