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Gini covariance matrix and its affine equivariant version

Author

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  • Xin Dang

    (University of Mississippi)

  • Hailin Sang

    (University of Mississippi)

  • Lauren Weatherall

    (BlueCross & BlueShield of Mississippi)

Abstract

We propose a new covariance matrix called Gini covariance matrix (GCM), which is a natural generalization of univariate Gini mean difference (GMD) to the multivariate case. The extension is based on the covariance representation of GMD by applying the multivariate spatial rank function. We study properties of GCM, especially in the elliptical distribution family. In order to gain the affine equivariance property for GCM, we utilize the transformation–retransformation (TR) technique and obtain an affine equivariant version GCM that turns out to be a symmetrized M-functional. The influence function of those two GCM’s are obtained and their estimation has been presented. Asymptotic results of estimators have been established. A closely related scatter Kotz functional and its estimator are also explored. Finally, asymptotical efficiency and finite sample efficiency of the TR version GCM are compared with those of sample covariance matrix, Tyler-M estimator and other scatter estimators under different distributions.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:stpapr:v:60:y:2019:i:3:d:10.1007_s00362-016-0842-z
    DOI: 10.1007/s00362-016-0842-z
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    References listed on IDEAS

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    Cited by:

    1. Xin Dang & Dao Nguyen & Yixin Chen & Junying Zhang, 2021. "A new Gini correlation between quantitative and qualitative variables," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(4), pages 1314-1343, December.
    2. Vanderford Courtney & Sang Yongli & Dang Xin, 2020. "Two symmetric and computationally efficient Gini correlations," Dependence Modeling, De Gruyter, vol. 8(1), pages 373-395, January.
    3. Vanderford Courtney & Sang Yongli & Dang Xin, 2020. "Two symmetric and computationally efficient Gini correlations," Dependence Modeling, De Gruyter, vol. 8(1), pages 373-395, January.

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