  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.
