Robustness of the Affine Equivariant Scatter Estimator Based on the Spatial Rank Covariance Matrix

Visuri et al. (2000 Visuri, S., Koivunen, V., Oja, H. (2000). Sign and rank covariance matrices. J. Stat. Plann. Inference 91:557–575.[Crossref], [Web of Science ®] , [Google Scholar]) proposed a technique for robust covariance matrix estimation based on different notions of multivariate sign and rank. Among them, the spatial rank based covariance matrix estimator that utilizes a robust scale estimator is especially appealing due to its high robustness, computational ease, and good efficiency. Also, it is orthogonally equivariant under any distribution and affinely equivariant under elliptically symmetric distributions. In this paper, we study robustness properties of the estimator with respective to two measures: breakdown point and influence function. More specifically, the upper bound of the finite sample breakdown point can be achieved by a proper choice of univariate robust scale estimator. The influence functions for eigenvalues and eigenvectors of the estimator are derived. They are found to be bounded under some assumptions. Moreover, finite sample efficiency comparisons to popular robust MCD, M, and S estimators are reported.