Inference on the shape of elliptical distributions based on the MCD

The minimum covariance determinant (MCD) estimator of scatter is one of the most famous robust procedures for multivariate scatter. Despite the quite important research activity related to this estimator, culminating in the recent thorough asymptotic study of Cator and Lopuhaä (2010, 2012), no results have been obtained on the corresponding estimator of shape, which is the parameter of interest in many multivariate problems (including principal component analysis, canonical correlation analysis, testing for sphericity, etc.) In this paper, we therefore propose and study MCD-based inference procedures for shape, that inherit the good robustness properties of the MCD. The main emphasis is on asymptotic results, for point estimation (Bahadur representation and asymptotic normality results) as well as for hypothesis testing (asymptotic distributions under the null and under local alternatives). Influence functions of the MCD-estimators of shape are obtained as a corollary. Monte-Carlo studies illustrate our asymptotic results and assess the robustness of the proposed procedures.