• Citer

Existence and breakdown analysis of M-quantiles in general Hilbert spaces

par Konen, Dimitri ;Paindaveine, Davy
Référence Electronic Journal of Statistics, 19, 2, page (5778-5804)
Publication Publié, 2025

Article révisé par les pairs

• Accès en ligne
• Détails
• Contenu
• Statistiques

Résumé :

M-quantiles extend M-estimators of location in the same way the usual quantiles extend the median. Since this extension is motivated by the desire to achieve a trade-off between robustness and efficiency, it is surprising that a robustness analysis of M-quantiles remains unavailable to date. Such an analysis is missing even for univariate M-quantiles, hence also for their most common multivariate extension, namely spatial or geometric M-quantiles. In this paper, we therefore study the global robustness of M-quantiles in terms of breakdown point. We do so in a general framework where M-quantiles, to the best of our knowledge, have not been considered earlier, namely in possibly infinite-dimensional Hilbert spaces (which covers the case of functional M-quantiles). Existence of M-quantiles in such a general framework remains an open question, though, and we thus first establish existence through weak topology arguments. Then, we study the breakdown point of Mquantiles. We provide an account of this question that requires only very mild assumptions on the convex loss function at hand. As a result, our analysis is considerably more general than the one, almost exclusively conducted for quantiles, in [20]. Such generality requires original results on regular variation of convex loss functions. In order to handle the possible non-uniqueness of M-quantiles, we also need to consider lower and upper breakdown point concepts.