The complexity of computing the MCD-estimator

In modern statistics the robust estimation of parameters is a central problem, i. e., an estimation that is not or only slightly affected by outliers in the data. The Minimum Covariance Determinant estimator (MCD) [8] is probably one of the most important robust estimators of location and scatter. The complexity of computing the MCD, however, was unknown and generally thought to be exponential even if the dimensionality of the data is fixed. Here we present a polynomial time algorithm for MCD for fixed dimension of the data. In contrast we show that computing the MCDestimator is NP-hard if the dimension varies.