Extreme rainfall modelling under climate change and proper scoring rules for extremes and inference

Abstract

Model development, model inference and model evaluation are three important cornerstones of statistical analysis. This thesis touches on all these through modelling extremes under climate change and evaluating extreme models using scoring rules, and by using scoring rules for statistical inference on spatial models. The findings are presented in three papers. In Paper I, a new statistical model is developed, that uses the connections between the generalised extreme value distribution and the generalised Pareto distribution to capture frequency changes in annual maxima. This allows using high-quality annual maxima data instead of less-well checked daily data to separately estimate trends in frequency and intensity. The model was applied to annual maximum data of Volume 10 of NOAA Atlas 14, showing that in the Northeastern US there are evidence that extreme rainfall events are occurring more often with rising temperature, but that there is little evidence that there are trends in the distribution of sizes of individual extreme rainfall events. Paper II introduces the concept of local weight-scale invariance which is a relaxation of local scale invariance for proper scoring rules. This relaxation is suitable for weighted scores that are for example useful when comparing extreme models. A weight-scale invariant version of the tail-weighted continuous ranked probability score is introduced and the properties of the different weighted scores were investigated. Finally, Paper III continues on the path of scoring rules, but instead uses scoring rules for statistical inference of spatial models. The proposed approach estimates parameters of spatial models by maximising the average leave-one-out cross-validation score (LOOS). The method results in fast computations for Gaussian models with sparse precision matrices and allows tailoring estimator's robustness to outliers and their sensitivity to spatial variations of uncertainty through the choice of the scoring rule which is used in the maximisation.