Efficient R-Estimation of Principal and Common Principal Components

Abstract

We propose rank-based estimators of principal components, both in the one-sample and, under the assumption of common principal components, in the m-sample cases. Those estimators are obtained via a rank-based version of Le Cam’s one-step method, combined with an estimation of cross-information quantities. Under arbitrary elliptical distributions with, in the m-sample case, possibly heterogeneous radial densities, those R-estimators remain root-n consistent and asymptotically normal, while achieving asymptotic efficiency under correctly specified radial densities. Contrary to their traditional counterparts computed from empirical covariances, they do not require any moment conditions. When based on Gaussian score functions, in the one-sample case, they uniformly dominate their classical competitors in the Pitman sense. Their AREs with respect to other robust procedures are quite high—up to 30, in the Gaussian case, with respect to minimum covariance determinant estimators. Their finite-sample performances are investigated via a Monte Carlo study.