Graphical modelling and the Mahalanobis distance

I consider the problem of estimating the Mahalanobis distance between multivariate normal populations when the population covariance matrix satisfies a graphical model. In addition to providing a clear understanding of the dependencies in a multivariate data set, the use of graphical models can reduce the variability of the estimated distances and improve inferences. I derive the asymptotic distribution of the estimated Mahalanobis distance under a general covariance model, which includes graphical models as a special case. Two examples are discussed.