Final report on DFG project "Statistical Inference for Time Series Extremes based on the Sliding Block Maxima Method"

Abstract

Extreme value statistics is concerned with the statistical analysis of data samples, often collected over time, at extreme levels. Key quantities of interest include quantiles or return levels that exceed the largest observed value, as well as probabilities and return periods of rare events that may not have occurred within the observed sample. Applications of extreme value statistics are widespread, including fields such as finance, insurance, and environmental science. A common approach for statistically analyzing such extreme events is the block maxima method. Recent case-by-case studies have shown that respective estimators can be improved by an approach called `the sliding block maxima method'. The aim of this project was to enhance the sliding block maxima toolbox and to deepen our understanding of the mathematical principles that underpin it. In particular, asymptotic theory was developed for general U-statistics based on sliding block maxima, and a novel bootstrap technique was introduced to assess estimation uncertainties. Overall, the findings of this project pave the way for more efficient and accurate analysis of extreme events compared to traditional methods.