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In extreme value theory and other related risk analysis fields, probabilityweighted moments (PWM) have been frequently used to estimate the parameters ofclassical extreme value distributions. This method-of-moment technique can beapplied when second moments are finite, a reasonable assumption in manyenvironmental domains like climatological and hydrological studies. Threeadvantages of PWM estimators can be put forward: their simple interpretations,their rapid numerical implementation and their close connection to thewell-studied class of U-statistics. Concerning the later, this connection leadsto precise asymptotic properties, but non asymptotic bounds have been lackingwhen off-the-shelf techniques (Chernoff method) cannot be applied, asexponential moment assumptions become unrealistic in many extreme valuesettings. In addition, large values analysis is not immune to the undesirableeffect of outliers, for example, defective readings in satellite measurementsor possible anomalies in climate model runs. Recently, the treatment ofoutliers has sparked some interest in extreme value theory, but results aboutfinite sample bounds in a robust extreme value theory context are yet to befound, in particular for PWMs or tail index estimators. In this work, wepropose a new class of robust PWM estimators, inspired by the median-of-meansframework of Devroye et al. (2016). This class of robust estimators is shown tosatisfy a sub-Gaussian inequality when the assumption of finite second momentsholds. Such non asymptotic bounds are also derived under the generalcontamination model. Our main proposition confirms theoretically a trade-offbetween efficiency and robustness. Our simulation study indicates that, whileclassical estimators of PWMs can be highly sensitive to outliers.

Robust and non asymptotic estimation of probability weighted moments with application to extreme value analysis