  The Minimum Covariance Determinant (MCD) approach robustly estimates the
location and scatter matrix using the subset of given size with lowest sample
covariance determinant. Its main drawback is that it cannot be applied when the
dimension exceeds the subset size. We propose the Minimum Regularized
Covariance Determinant (MRCD) approach, which differs from the MCD in that the
scatter matrix is a convex combination of a target matrix and the sample
covariance matrix of the subset. A data-driven procedure sets the weight of the
target matrix, so that the regularization is only used when needed. The MRCD
estimator is defined in any dimension, is well-conditioned by construction and
preserves the good robustness properties of the MCD. We prove that so-called
concentration steps can be performed to reduce the MRCD objective function, and
we exploit this fact to construct a fast algorithm. We verify the accuracy and
robustness of the MRCD estimator in a simulation study and illustrate its
practical use for outlier detection and regression analysis on real-life
high-dimensional data sets in chemistry and criminology.
