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The affine equivariant sign covariance matrix: asymptotic behavior and efficiencies

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  • Ollila, Esa
  • Oja, Hannu
  • Croux, Christophe

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.

Suggested Citation

  • Ollila, Esa & Oja, Hannu & Croux, Christophe, 2003. "The affine equivariant sign covariance matrix: asymptotic behavior and efficiencies," Journal of Multivariate Analysis, Elsevier, vol. 87(2), pages 328-355, November.
    • Handle: RePEc:eee:jmvana:v:87:y:2003:i:2:p:328-355
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    References listed on IDEAS

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    1. Möttönen, J. & Hettmansperger, T. P. & Oja, H. & Tienari, J., 1998. "On the Efficiency of Affine Invariant Multivariate Rank Tests," Journal of Multivariate Analysis, Elsevier, vol. 66(1), pages 118-132, July.
    2. Croux, Christophe & Haesbroeck, Gentiane, 1999. "Influence Function and Efficiency of the Minimum Covariance Determinant Scatter Matrix Estimator," Journal of Multivariate Analysis, Elsevier, vol. 71(2), pages 161-190, November.
    3. Hannu Oja, 1999. "Affine Invariant Multivariate Sign and Rank Tests and Corresponding Estimates: a Review," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 26(3), pages 319-343, September.
    4. Oja, Hannu, 1983. "Descriptive statistics for multivariate distributions," Statistics & Probability Letters, Elsevier, vol. 1(6), pages 327-332, October.
    5. Brown, Bruce M. & Hall, Peter & Young, G. Alastair, 1997. "On the Effect of Inliers on the Spatial Median," Journal of Multivariate Analysis, Elsevier, vol. 63(1), pages 88-104, October.
    6. Esa Ollila & Hannu Oja & Thomas P. Hettmansperger, 2002. "Estimates of regression coefficients based on the sign covariance matrix," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(3), pages 447-466, August.
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    Cited by:

    1. Xin Dang & Hailin Sang & Lauren Weatherall, 2019. "Gini covariance matrix and its affine equivariant version," Statistical Papers, Springer, vol. 60(3), pages 641-666, June.
    2. Hallin Marc & Paindaveine Davy, 2006. "Parametric and semiparametric inference for shape: the role of the scale functional," Statistics & Risk Modeling, De Gruyter, vol. 24(3), pages 327-350, December.

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