Statistical Papers

, Volume 60, Issue 3, pp 641–666 | Cite as

Gini covariance matrix and its affine equivariant version

  • Xin DangEmail author
  • Hailin Sang
  • Lauren Weatherall
Regular Article


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.


Affine equivariance Efficiency Gini mean difference Influence function Scatter M-estimator Spatial rank Symmetrization 

Mathematics Subject Classification

62H10 62H12 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MississippiUniversityUSA
  2. 2.BlueCross & BlueShield of MississippiFlowoodUSA

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