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Dang, Xin; Sang, Hailin; Weatherall, Lauren

Gini covariance matrix and its affine equivariant version. (English) Zbl 1419.62129

Stat. Pap. 60, No. 3, 291-316 (2019).
Summary: 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.

Cited in 3 Documents

MSC:

62H10 Multivariate distribution of statistics
62H12 Estimation in multivariate analysis
62G20 Asymptotic properties of nonparametric inference

Keywords:

affine equivariance; efficiency; Gini mean difference; influence function; scatter M-estimator; spatial rank; symmetrization

Software:

MNM; mnormt; ICSNP; fastM
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References:

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