Estimating return levels from maxima of non-stationary random sequences using the Generalized PWM method

Abstract. Since the pioneering work of Landwehr et al. (1979), Hosking et al. (1985) and their collaborators, the Probability Weighted Moments (PWM) method has been very popular, simple and efficient to estimate the parameters of the Generalized Extreme Value (GEV) distribution when modeling the distribution of maxima (e.g., annual maxima of precipitations) in the Identically and Independently Distributed (IID) context. When the IID assumption is not satisfied, a flexible alternative, the Maximum Likelihood Estimation (MLE) approach offers an elegant way to handle non-stationarities by letting the GEV parameters to be time dependent. Despite its qualities, the MLE applied to the GEV distribution does not always provide accurate return level estimates, especially for small sample sizes or heavy tails. These drawbacks are particularly true in some non-stationary situations. To reduce these negative effects, we propose to extend the PWM method to a more general framework that enables us to model temporal covariates and provide accurate GEV-based return levels. Theoretical properties of our estimators are discussed. Small and moderate sample sizes simulations in a non-stationary context are analyzed and two brief applications to annual maxima of CO₂ and seasonal maxima of cumulated daily precipitations are presented.