Asymptotic distributions of robust shape matrices and scales

It has been frequently observed in the literature that many multivariate statistical methods require the covariance or dispersion matrix [Sigma] of an elliptical distribution only up to some scaling constant. If the topic of interest is not the scale but only the shape of the elliptical distribution, it is not meaningful to focus on the asymptotic distribution of an estimator for [Sigma] or another matrix [Gamma][is proportional to][Sigma]. In the present work, robust estimators for the shape matrix and the associated scale are investigated. Explicit expressions for their joint asymptotic distributions are derived. It turns out that if the joint asymptotic distribution is normal, the estimators presented are asymptotically independent for one and only one specific choice of the scale function. If it is non-normal (this holds for example if the estimators for the shape matrix and scale are based on the minimum volume ellipsoid estimator) only the scale function presented leads to asymptotically uncorrelated estimators. This is a generalization of a result obtained by Paindaveine [D. Paindaveine, A canonical definition of shape, Statistics and Probability Letters 78 (2008) 2240-2247] in the context of local asymptotic normality theory.