Gini Covariance Matrix and its Affine Equivariant Version

TL;DR

This paper introduces the Gini covariance matrix (GCM), a new multivariate measure based on Gini mean difference, and develops an affine equivariant version with desirable statistical properties and efficiency comparisons.

Contribution

It proposes the Gini covariance matrix and its affine equivariant version, extending univariate Gini mean difference to multivariate data with theoretical and practical analysis.

Findings

GCM has desirable properties in elliptical distributions

Affine equivariant GCM is a symmetrized M-functional

TR version GCM shows competitive efficiency

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.