A note on finite-sample efficiencies of estimators for the minimum volume ellipsoid

Abstract

Among the most well known estimators of multivariate location and scatter is the Minimum Volume Ellipsoid (MVE). Many algorithms have been proposed to compute it. Most of these attempt merely to approximate as close as possible the exact MVE, but some of them led to the definition of new estimators which maintain the properties of robustness and affine equivariance that make the MVE so attractive. Rousseeuw and van Zomeren (1990) used the <$>(p+1)<$>- subset estimator which was modified by Croux and Haesbroeck (1997) to give rise to the averaged <$>(p+1)<$>- subset estimator . This note shows by means of simulations that the averaged <$>(p+1)<$>-subset estimator outperforms the exact estimator as far as finite-sample efficiency is concerned. We also present a new robust estimator for the MVE, closely related to the averaged <$>(p+1)<$>-subset estimator, but yielding a natural ranking of the data.