Intriguing Properties of Extreme Geometric Quantiles

Abstract

Central properties of geometric quantiles have been well-established in the recent statistical literature. In this study, we try to get a grasp of how extreme geometric quantiles behave. Their asymptotics are provided, both in direction and magnitude, under suitable moment conditions, when the norm of the associated index vector tends to one. Some intriguing properties are highlighted: in particular, it appears that if a random vector has a finite covariance matrix, then the magnitude of its extreme geometric quantiles grows at a fixed rate. We take profit of these results by defining a parametric estimator of extreme geometric quantiles of such a random vector. The consistency and asymptotic normality of the estimator are established, and contrasted with what can be obtained for univariate quantiles. Our results are illustrated on both simulated and real data sets. As a conclusion, we deduce from our observations some warnings which we believe should be known by practitioners who would like to use such a notion of multivariate quantile to detect outliers or analyze extremes of a random vector.