Small - sample correction factor of the minimum covariance determinant estimator

Abstract

The Minimum Covariance Determinant (MCD) estimator proposed by Rousseeuw (1985) is a highly robust way to estimate location and scatter of multivariate and possibly contaminated data sets. Recently Rousseeuw and Van Driessen (1999) have proposed a fast algorithm which allows practical application of MCD even in very large and contaminated data sets. As an initial estimate of location and scatter, MCD could be a better alternative than the so-called minimum volume ellipsoid (MVE) estimator proposed by Rousseeuw (1985), because MCD is asymptotically normal and MVE has a lower convergence rate. Robust distances using MVE have been used for labeling multiple outliers, as it was suggested by Rousseeuw and van Zomeren (1990). For small - sample cases Rousseeuw and van Zomeren (1991) computed by simulation a correction factor in terms to be able to compare MVE robust distances against distribution quantiles. In accordance with its superior theoretical performance, MCD estimate might be used for the same purpose; however it also requires a correction factor for small samples. We present in this paper the results from a simulation study for small - sample cases, with two objectives: first, when both types of estimators are used to build tolerance regions, we compare them in relation to their actual coverage probabilities; and second, to compute required correction factors. We followed the methodology from Rousseeuw and van Zomeren (1991).

Keywords:

• robust estimation
• outlier detection
• correction factor