University of Washington - Department of Statistics
This talk provides an introduction to robust estimation of covariance matrices, covering both theoretical and computational aspects, and indicating what we believe to be best choice of estimator at the present time. We begin with a brief introduction to the main concepts of robustness, focusing primarily on minimizing maximum bias for a class of standard multivariate mixture outlier generating models, while maintaining high efficiency at the nominal model. Then we introduce the main classes of robust estimates of covariance: M-estimates, estimates based on a robust scale measure (MVE, MCD and general S-estimates), projection estimates (Donoho-Stahel), and discuss efficient computational methods for these estimates. We also discuss estimates based on pair wise robust covariances that have attractive properties with respect to computational speed, and also with regard to effectiveness for a non-standard class of outlier generating models that is often more realistic than the standard models. We motivate the talk and illustrate some of the methods with examples from finance where covariances and correlations play a fundamental role in portfolio construction, and where frequently occurring outliers can have a quite adverse influence on classical portfolio construction methods. The main reference for this talk is Chapter 6 of the forthcoming book Robust Statistics: Theory and Methods by Maronna, Martin and Yohai, due in March 2005 from Wiley. Copies of the chapter will be provided at the talk.