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Asymptotics of the two-stage spatial sign correlation

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  • Dürre, Alexander
  • Vogel, Daniel

Abstract

The spatial sign correlation (Dürre et al., 2015) is a highly robust and easy-to-compute, bivariate correlation estimator based on the spatial sign covariance matrix. Since the estimator is inefficient when the marginal scales strongly differ, a two-stage version was proposed. In the first step, the observations are marginally standardized by means of a robust scale estimator, and in the second step, the spatial sign correlation of the thus transformed data set is computed. Dürre et al. (2015) give some evidence that the asymptotic distribution of the two-stage estimator equals that of the spatial sign correlation at equal marginal scales by comparing their influence functions and presenting simulation results, but give no formal proof. In the present paper, we close this gap and establish the asymptotic normality of the two-stage spatial sign correlation and compute its asymptotic variance for elliptical population distributions. We further derive a variance-stabilizing transformation in the same vein a Fisher’s z-transform. This variance-stabilizing transform is valid for all elliptical distributions and yields very accurate confidence intervals.

Suggested Citation

  • Dürre, Alexander & Vogel, Daniel, 2016. "Asymptotics of the two-stage spatial sign correlation," Journal of Multivariate Analysis, Elsevier, vol. 144(C), pages 54-67.
    • Handle: RePEc:eee:jmvana:v:144:y:2016:i:c:p:54-67
    DOI: 10.1016/j.jmva.2015.10.011
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    References listed on IDEAS

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    1. D. Vogel & R. Fried, 2011. "Elliptical graphical modelling," Biometrika, Biometrika Trust, vol. 98(4), pages 935-951.
    2. N. Locantore & J. Marron & D. Simpson & N. Tripoli & J. Zhang & K. Cohen & Graciela Boente & Ricardo Fraiman & Babette Brumback & Christophe Croux & Jianqing Fan & Alois Kneip & John Marden & Daniel P, 1999. "Robust principal component analysis for functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 8(1), pages 1-73, June.
    3. Daniel Gervini, 2008. "Robust functional estimation using the median and spherical principal components," Biometrika, Biometrika Trust, vol. 95(3), pages 587-600.
    4. Dürre, Alexander & Vogel, Daniel & Tyler, David E., 2014. "The spatial sign covariance matrix with unknown location," Journal of Multivariate Analysis, Elsevier, vol. 130(C), pages 107-117.
    5. Frahm, Gabriel, 2009. "Asymptotic distributions of robust shape matrices and scales," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1329-1337, August.
    6. C. Croux & C. Dehon & A. Yadine, 2010. "The k-step spatial sign covariance matrix," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 4(2), pages 137-150, September.
    7. Marden, John I., 1999. "Some robust estimates of principal components," Statistics & Probability Letters, Elsevier, vol. 43(4), pages 349-359, July.
    8. Paindaveine, Davy, 2008. "A canonical definition of shape," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2240-2247, October.
    9. Wendler, Martin, 2011. "Bahadur representation for U-quantiles of dependent data," Journal of Multivariate Analysis, Elsevier, vol. 102(6), pages 1064-1079, July.
    10. Dürre, Alexander & Vogel, Daniel & Fried, Roland, 2015. "Spatial sign correlation," Journal of Multivariate Analysis, Elsevier, vol. 135(C), pages 89-105.
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    Cited by:

    1. Dürre, Alexander & Tyler, David E. & Vogel, Daniel, 2016. "On the eigenvalues of the spatial sign covariance matrix in more than two dimensions," Statistics & Probability Letters, Elsevier, vol. 111(C), pages 80-85.
    2. Raymaekers, Jakob & Rousseeuw, Peter, 2019. "A generalized spatial sign covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 94-111.

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