High breakdown estimators for principal components: the projection-pursuit approach revisited

Authors:
Christophe Croux;Anne Ruiz-Gazen
Affiliations:
Department of Applied Economics, K.U. Leuven, Naamsestraat 69, B-3000 Leuven, Belgium;GREMAQ (U.M.R. CNRS 5604), University Toulouse I, Manufacture des Tabacs, 21, al. de Brienne 31042 Toulouse Cedex, France and L.S.P. (U.M.R. CNRS 5583), University Toulouse III, 118 route de Narbo ...
Venue:
Journal of Multivariate Analysis
Year:
2005

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Abstract

Li and Chen (J. Amer. Statist. Assoc. 80 (1985) 759) proposed a method for principal components using projection-pursuit techniques. In classical principal components one searches for directions with maximal variance, and their approach consists of replacing this variance by a robust scale measure. Li and Chen showed that this estimator is consistent, qualitative robust and inherits the breakdown point of the robust scale estimator. We complete their study by deriving the influence function of the estimators for the eigenvectors, eigenvalues and the associated dispersion matrix. Corresponding Gaussian efficiencies are presented as well. Asymptotic normality of the estimators has been treated in a paper of Cui et al. (Biometrika 90 (2003) 953), complementing the results of this paper. Furthermore, a simple explicit version of the projection-pursuit based estimator is proposed and shown to be fast to compute, orthogonally equivariant, and having the maximal finite-sample breakdown point property. We will illustrate the method with a real data example.