  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.
