On quantiles, continuity and robustness
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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-Estimates (QoE) estimator, a robustification procedure for 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.
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