On quantiles, continuity and robustness

We consider the geometric quantile and various definitions of the component-wise quantile in infinite dimensions and show their existence, uniqueness and continuity. Building on these results, we introduce and study the properties of the quantile-of-estimators (QoE) estimator, a robustification of a large class of estimators. For example, given an estimator that is asymptotically normal, the QoE estimator is asymptotically normal even in the presence of contaminated data.