Minimum Covariance Determinant and Extensions

The Minimum Covariance Determinant (MCD) method is a highly robust estimator of multivariate location and scatter, for which a fast algorithm is available. Since estimating the covariance matrix is the cornerstone of many multivariate statistical methods, the MCD is an important building block when developing robust multivariate techniques. It also serves as a convenient and efficient tool for outlier detection. The MCD estimator is reviewed, along with its main properties such as affine equivariance, breakdown value, and influence function. We discuss its computation, and list applications and extensions of the MCD in applied and methodological multivariate statistics. Two recent extensions of the MCD are described. The first one is a fast deterministic algorithm which inherits the robustness of the MCD while being almost affine equivariant. The second is tailored to high-dimensional data, possibly with more dimensions than cases, and incorporates regularization to prevent singular matrices.