Some negative results on extreme multivariate quantiles defined through convex optimisation
Résumé
A discussion of some general properties that a notion of extreme multivariate quantile should satisfy will be given. We will then recall the concept of geometric quantile by transposing the definition of a univariate quantile as a minimiser of a cost function based on the so-called check function to the multivariate case. We shall then argue that extreme versions of these geometric quantiles are not suitable for the extreme-value analysis of a multivariate data set. A particular reason for this is that when the underlying distribution possesses a finite covariance matrix then the magnitude of these quantiles grows at a fixed rate that is independent of the distribution. We shall also discuss an extension of this negative result to the wider class of geometric M-quantiles.