Thomas Holgersson: Expected and unexpected values of individual Mahalanobis distances

We discuss the problem of estimating individual Mahalanobis distances (MD) in cases when the dimension of the variable p is proportional to the sample size n. Asymptotic expected values of the MD's are derived under the assumption p/n → c, 0 ≤ c < 1 for both the classical and the leave-one-out estimators. It is shown that some estimators are asymptotically biased but that biased-corrected versions are available. Moreover, a risk function is derived for finding an optimal estimate of the inverse covariance matrix on which the MD depends. This is then used to derive an estimator which, unlike the standard estimator, yields efficient MD estimates over the whole range 0 ≤ c < 1.