The affine equivariant sign covariance matrix: asymptotic behavior and efficiencies

Authors:
Esa Ollila;Hannu Oja;Christophe Croux
Affiliations:
Signal Processing Laboratory, Helsinki University of Technology, P.O. Box 3000, Espoo FIN-02015, Finland;University of Jyväskylä, Jyväskylä, Finland;University of Leuven, Leuven, Belgium
Venue:
Journal of Multivariate Analysis
Year:
2003

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Abstract

We consider the affine equivariant sign covariance matrix (SCM) introduced by Visuri et al. (J. Statist. Plann. Inference 91 (2000) 557). The population SCM is shown to be proportional to the inverse of the regular covariance matrix. The eigenvectors and standardized eigenvalues of the covariance matrix can thus be derived from the SCM. We also construct an estimate of the covariance and correlation matrix based on the SCM. The influence functions and limiting distributions of the SCM and its eigenvectors and eigenvalues are found. Limiting efficiencies are given in multivariate normal and t-distribution cases. The estimates are highly efficient in the multivariate normal case and perform better than estimates based on the sample covariance matrix for heavy-tailed distributions. Simulations confirmed these findings for finite-sample efficiencies.