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The affine equivariant sign covariance matrix: asymptotic behavior and efficiencies.

By E Ollila, H Oja and Christophe Croux

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. (C) 2003 Elsevier Science (USA). All rights reserved.affine equivariance; covariance and correlation matrices; efficiency; eigenvectors and eigenvalues; influence function; multivariate median; multivariate sign; robustness; multivariate location; discriminant-analysis; principal components; dispersion matrices; tests; estimators;

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Citations

  1. (1984). An Introduction to Multivariate Statistical Analysis. Second edition.
  2. (1994). Ane invariant multivariate multisample sign tests.
  3. (1994). Ane invariant multivariate one-sample sign tests.
  4. (1999). Ane invariant multivariate sign and rank tests and corresponding estimates: a review.
  5. (1980). Approximation Theorems of Mathematical Statistics.
  6. (1987). Asymptotic behavior of S-estimators of multivariate location parameters and dispersion matrices.
  7. (1983). Descriptive statistics for multivariate distributions.
  8. (2001). Estimates of regression coecients based on the sign covariance matrix. Conditionally accepted.
  9. (1999). In uence function and eciency of the minimum covariance determinant scatter matrix estimator.
  10. (1985). In uence in principal component analysis.
  11. (1988). Matrix Dierential Calculus with Applications in Statistics and Econometrics.
  12. (1985). Multivariate estimation with high breakdown point.
  13. (1998). On the eciency of ane invariant multivariate rank tests.
  14. (1997). On the eect of inliers on the spatial median.
  15. (1989). On the relation between S-estimators and M-estimators of multivariate location and covariance.
  16. (2000). Principal component analysis based on robust estimators of the covariance or correlation matrix : in uence functions and eciencies.
  17. (1996). Regularized gaussian discriminant analysis through eigenvalue decomposition.
  18. (1981). Robust estimation of dispersion matrices and principal components.
  19. (1998). Robust estimation of multivariate location and scatter.
  20. (2001). Robust linear discriminant analysis using S-estimators.
  21. (1986). Robust Statistics: The Approach Based on In uence Functions.
  22. (2000). Sign and rank covariance matrices.