Confidence Intervals for the Mahalanobis Distance

Mahalanobis distances appear, often in a disguised form, in many statistical problems dealing with comparing two multivariate normal populations. Assuming a common covariance matrix the overlapping coefficient (Bradley, E.L. Encyclopedia of Statistical Sciences; 1985), optimal error rates (Rao, P.S.R.S.; Dorvlo, A.S.S. Commun. Statist.—Simula. Computation 1985, 14, 774–790), and the generalized ROC criterion (Reiser, B.; Faraggi, D. Biometrics 1997, 53, 644–652) are all monotonic functions of the Mahalanobis distance. Approximate confidence intervals for all of these have appeared in the literature on an ad-hoc basis. In this paper, we provide a unified approach to obtaining an effectively exact confidence interval for the Mahalanobis distance and all the above measures.