Asymptotic expansion of the minimum covariance determinant estimators

In arXiv:0907.0079 by Cator and Lopuhaa, an asymptotic expansion for the MCD estimators is established in a very general framework. This expansion requires the existence and non-singularity of the derivative in a first-order Taylor expansion. In this paper, we prove the existence of this derivative for multivariate distributions that have a density and provide an explicit expression. Moreover, under suitable symmetry conditions on the density, we show that this derivative is non-singular. These symmetry conditions include the elliptically contoured multivariate location-scatter model, in which case we show that the minimum covariance determinant (MCD) estimators of multivariate location and covariance are asymptotically equivalent to a sum of independent identically distributed vector and matrix valued random elements, respectively. This provides a proof of asymptotic normality and a precise description of the limiting covariance structure for the MCD estimators.