Expected and unexpected values of individual Mahalanobis distances
Deliang Dai, Thomas Holgersson, Peter S. Karlsson · Communication in Statistics- Theory and Methods · 2016
This paper derives first-order sampling moments of individual Mahalanobis distances (MDs) in cases when the dimension p of the variable is proportional to the sample size n. Asymptotic expected values when n, p → ∞ are derived under the assumption p/n → c, 0 ⩽ c < 1. It is shown that some types of standard estimators remain unbiased in this case, while others are asymptotically biased, a property that appears to be unnoticed in the literature. Second-order moments are also supplied to give some additional insight to the matter.