Severe weather events, for instance, heavy rainfall, snow-melt or droughts, cause large losses of lives and money every year. Insurance companies offer some form of protection against such undesirable outcomes, and decision makers want to take precautions to prevent future catastrophes. Both, decision makers and insurance companies, are hence interested to understand which weather events induce a high risk. This information then allows the insurance companies to set premiums for their policies by predicting future losses. Further, the relationship between damages and weather is also important to assess the impact of climate change.
Several aspects have to be considered in the statistical modelling of this relationship. For instance, some regions in the world are more used to severe rainfall events than others and, hence, presumably less vulnerable to small amounts of rainfall than others. Spatial statistics provides a statistical framework which allows for a spatially varying relationship while accounting for certain similarities for areas which are geographically close. Further, damages, especially large losses, are rather rare and the statistical analysis is hence usually based on a low number of observations. Methods from extreme value theory consider the modelling of such events and may hence be beneficial.
This thesis aims to develop statistical models for the relationship between damages, in particular property insurance claims, and weather events, based on daily Norwegian insurance and weather data. To improve existing models, new methodology is introduced which allows for substantial flexibility of the statistical model. The risk induced by certain weather events is assumed to be spatially varying across Norway but with neighbouring regions exhibiting similar vulnerability. To account for certain non-linear effects, the class of monotonic regression functions is considered. Specifically, this work is the first to de- fine flexible dependence structures for such functions. In particular, the first approach considers a Bayesian framework and estimates are obtained by Markov chain Monte Carlo algorithms while the second approach is optimization-based. The last part of the thesis derives extreme value models for discrete data and estimates them in a Bayesian framework. In particular, a mixture model which allows for a flexible tail behaviour is motivated by an exploratory analysis of the highest claims in the data. Additionally, the data are restructured based on spatial and temporal patterns and then combined with the proposed extreme value mixture model. All these approaches, monotonic regression and extreme value analysis, lead to an improved model fit and a better understanding of the relationship between insurance claims and weather events.