Computable Robust Estimation of Multivariate Location and Shape in High Dimension Using Compound Estimators

Abstract

Estimation of multivariate shape and location in a fashion that is robust with respect to outliers and is affine equivariant represents a significant challenge. The use of compound estimators that use a combinatorial estimator such as Rousseeuw's minimum volume ellipsoid (MVE) or minimum covariance determinant (MCD) to find good starting points for high-efficiency robust estimators such as S estimators has been proposed. In this article we indicate why this scheme will fail in high dimension due to combinatorial explosion in the space that must be searched for the MVE or MCD. We propose a meta-algorithm based on partitioning the data that enables compound estimators to work in high dimension. We show that even when the computational effort is restricted to a linear function of the number of data points, the algorithm results in an estimator with good asymptotic properties. Extensive computational experiments are used to confirm that significant benefits accrue in finite samples as well. We also give empirical results indicating that the MCD is preferred over the MVE for this application.

Key Words:

• Minimum covariance determinant estimator
• Minimum volume ellipsoid estimator
• S estimator