Robust and non asymptotic estimation of probability weighted moments with application to extreme value analysis
摘要
In extreme value theory and other related risk analysis fields, probability weighted moments (PWM) have been frequently used to estimate the parameters of classical extreme value distributions, in situations when only the first moments exist, such as in many environmental domains like climatological and hydrological studies. Three advantages of PWM estimators can be put forward: their simple interpretations, their rapid numerical implementation and their close connection to the well-studied class of U-statistics. Concerning the latter, this connection leads to precise asymptotic properties, but non asymptotic bounds have been lacking when off-the-shelf techniques (Chernoff method) cannot be applied, as exponential moment assumptions become unrealistic in many extreme value settings. In addition, large values analysis is not immune to the undesirable effect of outliers, for example, defective readings in satellite measurements or possible anomalies in climate model runs. Recently, the treatment of outliers has sparked some interest in extreme value theory, but results about finite sample bounds in a robust extreme value theory context are yet to be found, in particular for PWMs or tail index estimators. In this work, we propose a new class of robust PWM estimators, inspired by the median-of-means framework. Focusing mainly on the i.i.d. case, this class of estimators is shown to satisfy a sub-Gaussian inequality under the only assumption that second moments are finite. Such non asymptotic bounds are also derived under the general contamination model. Our simulation study indicates that, while classical estimators of PWMs can be highly sensitive to outliers, our new approach remains weakly affected by the degree contamination.