Résumé : In hydrology and other applied fields, Probability Weighted Moments (PWM) have been frequently used to estimate the parameters of classical extreme value distributions. This method-of-moments technique can be applied when second moments are finite, a reasonable assumption in hydrology. Two advantages of PWM estimators are their ease of implementation and their close connection to the well-studied class of U-statistics. Consequently, precise asymptotic properties can be deduced. In practice, sample sizes are always finite and, depending on the application at hand, the sample length can be small, e.g. a sample of only 30 years of daily precipitation is quite common in some regions of the globe. In such a context, asymptotic theory is on a shaky ground and it is desirable to get non-asymptotic bounds.
Deriving such bounds from off-the-shelf techniques (Chernoff method) requires exponential moment assumptions, which are unrealistic in many settings. To bypass this hurdle, we propose a new estimator for PWM, inspired by the median-of-means framework of Devroye-Lerasle-Lugosi-Oliveira (AoS 2016). This estimator is then shown to satisfy a sub-Gaussian inequality, with only second moment assumptions. This allows us to derive non-asymptotic bounds for the estimation of the parameters of extreme value distributions and of extreme quantiles.
Travail en collaboration avec Anna Ben-Hamou et Philippe Naveau.