Statistica Sinica 14(2004), 297-316

INFLUENCE FUNCTION AND ASYMPTOTIC EFFICIENCY

OF THE AFFINE EQUIVARIANT RANK

COVARIANCE MATRIX

Esa Ollila, Christophe Croux and Hannu Oja

Helsinki University of Technology, University of Leuven

and University of Jyväskylä

Abstract: Visuri, Koivunen and Oja (2003) proposed and illustrated the use of the affine equivariant rank covariance matrix (RCM) in classical multivariate inference problems. The RCM was shown to be asymptotically multinormal but explicit formulas for the limiting variances and covariances were not given. In this paper the influence functions and the limiting variances and covariances of the RCM and the corresponding scatter estimate are derived in the multivariate elliptical case. Limiting efficiencies are given in the multivariate normal and $t$ distribution cases. The estimates based on the RCM are highly efficient in the multinormal case, and for heavy-tailed distribution, perform better than those based on the regular covariance matrix. Finite-sample and asymptotic efficiency comparisons to a selected redecending $M$-estimator and $S$-estimator are reported.

Key words and phrases: Elliptical distribution, influence function, multivariate analysis, multivariate rank, scatter matrix.