Generalized radius processes for elliptically contoured distributions

The use of Mahalanobis distances has a long history in statistics. Given a sample of size n and general location and scatter estimators, m(n) and Sigma(n), we can define "generalized" radii as r(i)(n) = root(X-i-m(n))' Sigma(-1)(n) (X-i-m(n)). If we wish to trim observations based on the estimators m(n) and Sigma(n), then it is natural to first remove the most remote ones (i.e., those with the largest r(i)(n,)s). With this in mind, we define a process that maps the trimming proportion, alpha in [0, 1], to the generalized radius of the observation that has just been removed by this level of trimming. We analyze the asymptotic behavior of this process for elliptically contoured distributions. We show that the limit law depends only on the elliptical family considered and how Sigma(n) serves to estimate the underlying "scale" factor through its determinant. We carry out Monte Carlo simulations for finite sample sizes, and outline an application for assessing fit to a fixed elliptical family and also for the case where a proportion of outlying observations is discarded.