Wildfires in the United States present a complexity of problems for private landowners and policy makers. This thesis takes a look at two key issues faced by private and government stakeholders; the first being a lack of knowledge regarding current prescribed fire laws and regulations. A legal review of administrative laws and regulations for prescribed burning in the Southeastern United States in the context of management-based regulation is used to address this issue. It was found that regulation for prescribed burning has shifted to a more management–based regime. The second is an empirical study of wildfire distribution in the state of Mississippi. Wildfires appear to fit a Pareto distribution throughout the state given a certain threshold. When analyzed in conjunction both studies could aid lawmakers in projecting the effects of a given policy change on actual wildfire occurrences and distribution.

The peaks over threshold (POT) method provides an attractive framework for estimating the risk of extreme events such as severe storms or large insurance claims. However, the conventional POT procedure, where the threshold excesses are modelled by a generalized Pareto distribution, suffers from small samples and subjective threshold selection. In recent years, two alternative approaches have been proposed in the form of mixture models that estimate the threshold and a folding procedure that generates larger tail samples. In this paper the empirical performances of the conventional POT procedure, the folding procedure and a mixture model are compared by modelling data sets on fire insurance claims and hurricane damage costs. The results show that the folding procedure gives smaller standard errors of the parameter estimates and in some cases more stable quantile estimates than the conventional POT procedure. The mixture model estimates are dependent on the starting values in the numerical maximum likelihood estimation, and are therefore difficult to compare with those from the other procedures. The conclusion is that none of the procedures is overall better than the others but that there are situations where one method may be preferred.

Title: Large claims modeling Author: Barbora Zuzáková Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Michal Pešta, Ph.D. Abstract: This thesis discusses a statistical modeling approach based on the extreme value theory to describe the behaviour of large claims of an insurance portfolio. We focus on threshold models which analyze exceedances of a high threshold. This approach has gained in popularity in recent years, as compared with the much older methods based directly on the extreme value distributions. The method is illustated using the group medical claims database recorded over the periods 1997, 1998 and 1999 maintained by the Society of Actuaries. We aim to demonstrate that the proposed model outperforms classical parametric distri- butions and thus enables to estimate high quantiles or the probable maximum loss more precisely. Keywords: threshold models, generalized Pareto distribution, large claims. 1

A teoria dos valores extremos é um tópico da probabilidade que descreve a distribuição assintótica das estatísticas de ordem, tais como máximos ou mínimos, de uma seqüência de variáveis aleatórias que seguem uma função de distribuição F normalmente desconhecida. Descreve, ainda, a distribuição assintótica dos excessos acima de um valor limiar de um ou mais termos dessa seqüência. Dessa forma, as metodologias padrões utilizada neste contexto consistem no ajuste da distribuição generalizada dos valores extremos a uma série de máximos anuais ou no ajuste da distribuição generalizada de Pareto a uma série de dados compostas somente de observações excedentes de um valor limiar. No entanto, segundo Coles et al. (2003), há uma crescente insatisfação com o desempenho destes modelos padrões para predição de eventos extremos causada, possivelmente, por pressuposições não atendidas como a de independência das observações ou pelo fato de que os mesmos não sejam recomendados para serem utilizados em algumas situações específicas como por exemplo e quando observações de máximos anuais compostas por duas ou mais populações independentes de eventos extremos sendo que a primeira descreve eventos menos freqüentes e de maior magnitude e a segunda descreve eventos mais freqüentes e de menor magnitude. Então, os dois artigos que compõem este trabalho tem como objetivo apresentar alternativas de análise de valores extremos para estas situações em que o ajuste dos modelos padrões não são adequados. No primeiro, foram ajustadas as distribuições generalizada de Pareto e exponencial, caso particular da GP, aos dados de vazão média diária do Posto de Artemis, Piracicaba, SP, Brasil, conjuntamente com a técnica do desagrupamento, (declustering), e comparadas as estimativas dos níveis de retorno para períodos de 5, 10, 50 e 100 anos. Conclui-se que as estimativas intervalares dos níveis de retorno obtidas por meio do ajuste da distribuição exponencial são mais precisas do que as obtidas com o ajuste da distribuição generalizada de Pareto. No segundo artigo, por sua vez, foi apresentada uma metodologia para o ajuste da distribuição de Gumbel e de misturas de duas distribuições de Gumbel aos dados de velocidades de ventos mensais de Piracicaba, SP. Selecionou-se a distribuição que melhor ajustou-se aos dados por meio de testes de hipóteses bootstrap paramétrico e critérios de seleção AIC e BIC. E concluiu-se que a mistura de duas distribuições de Gumbel é a distribuição que melhor se ajustou-se aos dados de velocidades máxima de ventos dos meses de abril e maio, enquanto que o ajuste da distribuição de Gumbel foi o melhor para os meses de agosto e setembro. / The extreme value theory is a probability topics that describes the asymtoptic distribution of order statistics such as maximum or minimum of random variables sequence that follow a distribution function F normaly unknown. Describes still, the excess asymtoptic distribution over threshold of this sequence. So, the standard methodologies of extremes values analysis are the fitting of generalized extreme value distribution to yearly maximum series or the fitting of generalized Pareto distribution to partial duration series. However, according to Coles et al. (2003), there is a growing dissatisfaction with the use this standard models for the prediction of extremes events and one of possible causes this fact may be a false assumptions about a sequence of observed data as a independence assumptions or because the standards models must not used in some specific situations like for example when maximum sample arise from two or more independents populations, where the first population describes more frequents and low intense events and the second population describes less frequents and more intense events. In this way, the two articles this work has a objective show alternatives about extreme values analysis for this situations that the standards models doesn´t recommended. In the first article, the generalized distribution Pareto and exponencial distribution, particular case of GP, together with to declustering methods was applied to mean daily flow of the Piracicaba river, Artemis station, Piracicaba, SP, and the estimates the return levels of 5, 10, 50 and 100 years were compared. We conclude that the interval estimates of the 50 and 100 year return levels obtained using the fitting the exponencial distribution are more precise than those obtained using the generalized Pareto distribution. In the second article, we propose the fit of Gumbel distribution and the Gumbel mixture to data maximum speed wind in Piracicaba, SP. We select the best model using bootstrap test of hypotheses and the AIC and BIC selection criteria We conclude that the mixture Gumbel is the best model to analyze the maximum wind speed data for months of april e may and otherside the fit of Gumbel distributions was the best fit to months of august e september.

A teoria dos valores extremos é um tópico da probabilidade que descreve a distribuição assintótica das estatísticas de ordem, tais como máximos ou mínimos, de uma seqüência de variáveis aleatórias que seguem uma função de distribuição F normalmente desconhecida. Descreve, ainda, a distribuição assintótica dos excessos acima de um valor limiar de um ou mais termos dessa seqüência. Dessa forma, as metodologias padrões utilizada neste contexto consistem no ajuste da distribuição generalizada dos valores extremos a uma série de máximos anuais ou no ajuste da distribuição generalizada de Pareto a uma série de dados compostas somente de observações excedentes de um valor limiar. No entanto, segundo Coles et al. (2003), há uma crescente insatisfação com o desempenho destes modelos padrões para predição de eventos extremos causada, possivelmente, por pressuposições não atendidas como a de independência das observações ou pelo fato de que os mesmos não sejam recomendados para serem utilizados em algumas situações específicas como por exemplo e quando observações de máximos anuais compostas por duas ou mais populações independentes de eventos extremos sendo que a primeira descreve eventos menos freqüentes e de maior magnitude e a segunda descreve eventos mais freqüentes e de menor magnitude. Então, os dois artigos que compõem este trabalho tem como objetivo apresentar alternativas de análise de valores extremos para estas situações em que o ajuste dos modelos padrões não são adequados. No primeiro, foram ajustadas as distribuições generalizada de Pareto e exponencial, caso particular da GP, aos dados de vazão média diária do Posto de Artemis, Piracicaba, SP, Brasil, conjuntamente com a técnica do desagrupamento, (declustering), e comparadas as estimativas dos níveis de retorno para períodos de 5, 10, 50 e 100 anos. Conclui-se que as estimativas intervalares dos níveis de retorno obtidas por meio do ajuste da distribuição exponencial são mais precisas do que as obtidas com o ajuste da distribuição generalizada de Pareto. No segundo artigo, por sua vez, foi apresentada uma metodologia para o ajuste da distribuição de Gumbel e de misturas de duas distribuições de Gumbel aos dados de velocidades de ventos mensais de Piracicaba, SP. Selecionou-se a distribuição que melhor ajustou-se aos dados por meio de testes de hipóteses bootstrap paramétrico e critérios de seleção AIC e BIC. E concluiu-se que a mistura de duas distribuições de Gumbel é a distribuição que melhor se ajustou-se aos dados de velocidades máxima de ventos dos meses de abril e maio, enquanto que o ajuste da distribuição de Gumbel foi o melhor para os meses de agosto e setembro. / The extreme value theory is a probability topics that describes the asymtoptic distribution of order statistics such as maximum or minimum of random variables sequence that follow a distribution function F normaly unknown. Describes still, the excess asymtoptic distribution over threshold of this sequence. So, the standard methodologies of extremes values analysis are the fitting of generalized extreme value distribution to yearly maximum series or the fitting of generalized Pareto distribution to partial duration series. However, according to Coles et al. (2003), there is a growing dissatisfaction with the use this standard models for the prediction of extremes events and one of possible causes this fact may be a false assumptions about a sequence of observed data as a independence assumptions or because the standards models must not used in some specific situations like for example when maximum sample arise from two or more independents populations, where the first population describes more frequents and low intense events and the second population describes less frequents and more intense events. In this way, the two articles this work has a objective show alternatives about extreme values analysis for this situations that the standards models doesn´t recommended. In the first article, the generalized distribution Pareto and exponencial distribution, particular case of GP, together with to declustering methods was applied to mean daily flow of the Piracicaba river, Artemis station, Piracicaba, SP, and the estimates the return levels of 5, 10, 50 and 100 years were compared. We conclude that the interval estimates of the 50 and 100 year return levels obtained using the fitting the exponencial distribution are more precise than those obtained using the generalized Pareto distribution. In the second article, we propose the fit of Gumbel distribution and the Gumbel mixture to data maximum speed wind in Piracicaba, SP. We select the best model using bootstrap test of hypotheses and the AIC and BIC selection criteria We conclude that the mixture Gumbel is the best model to analyze the maximum wind speed data for months of april e may and otherside the fit of Gumbel distributions was the best fit to months of august e september.

In many environmental and climate applications, extreme data are spatial by nature, and hence statistics of spatial extremes is currently an important and active area of research dedicated to developing innovative and flexible statistical models that determine the location, intensity, and magnitude of extreme events. In particular, the development of flexible sub-asymptotic models is in trend due to their flexibility in modeling spatial high threshold exceedances in larger spatial dimensions and with little or no effects on the choice of threshold, which is complicated with classical extreme value processes, such as Pareto processes. In this thesis, we develop new flexible sub-asymptotic extreme value models for modeling spatial and spatio-temporal extremes that are combined with carefully designed gradient-based Markov chain Monte Carlo (MCMC) sampling schemes and that can be exploited to address important scientific questions related to risk assessment in a wide range of environmental applications. The methodological developments are centered around two distinct themes, namely (i) sub-asymptotic Bayesian models for extremes; and (ii) flexible marked point process models with sub-asymptotic marks. In the first part, we develop several types of new flexible models for light-tailed and heavy-tailed data, which extend a hierarchical representation of the classical generalized Pareto (GP) limit for threshold exceedances. Spatial dependence is modeled through latent processes. We study the theoretical properties of our new methodology and demonstrate it by simulation and applications to precipitation extremes in both Germany and Spain. In the second part, we construct new marked point process models, where interest mostly lies in the extremes of the mark distribution. Our proposed joint models exploit intrinsic CAR priors to capture the spatial effects in landslide counts and sizes, while the mark distribution is assumed to take various parametric forms. We demonstrate that having a sub-asymptotic distribution for landslide sizes provides extra flexibility to accurately capture small to large and especially extreme, devastating landslides.

Title: Large deviations and their applications in insurance mathematics Author: Lucia Fuchsová Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Zbyněk Pawlas, Ph.D. Supervisor's e-mail address: Zbynek.Pawlas@mff.cuni.cz Abstract: In the present work we study large deviations theory. We discuss heavy-tailed distributions, which describe the probability of large claim oc- curence. We are interested in the use of large deviations theory in insurance. We simulate claim sizes and their arrival times for Cramér-Lundberg model and first we analyze the probability that ruin happens in dependence on the parameters of our model for Pareto distributed claim size, next we compare ruin probability for other claim size distributions. For real life data we model the probability of large claim size occurence by generalized Pareto distribu- tion. 1

In the present work we deal with the problem of etremal value of time series, especially of maxima. We study times and values of maximum by an approach of point process and we model distribution of extremal values by statistical methods. We estimate parameters of distribution using different methods, namely graphical methods of data analysis and subsequently we test the estimated distribution by tests of goodness of fit. We study the stationary case and also the cases with a trend. In connection with distribution of excesess and exceedances over a threshold we deal with generalized Pareto distribution.

This thesis is focused on the models based on extreme value theory and their practical applications. Specifically are described the block maxima models and the models based on threshold exceedances. Both of these methods are described in thesis theoretically. Apart from theoretical description there are also practical calculations based on simulated or real data. The applications of block maxima models are focused on choice of block size, suitability of the models for specific data and possibilities of extreme data analysis. The applications of models based on threshold exceedances are focused on choice of threshold and on suitability of the models. There is an example of the model used for calculations of reinsurance premium for extreme claims in the case of nonproportional reinsurance.

In the present thesis we will firstly familiarize ourselves with the term of operational risk, it's definition presented in the directives Basel II and Solvency II, and afterwards with the methods of calculation Capital Requirements for Operational Risk, set by these directives. In the second part of the thesis we will concentrate on the methods of modelling operational loss data. We will introduce the Extreme Value Theory which describes possible approaches to modelling data with significant values that occur infrequently; the typical characteristic of operational risk data. We will mainly focus on the model for threshold exceedances which utilizes Generalized Pareto Distribution to model the distribution of those excesses. The teoretical knowledge of this theory and the appropriate modelling will be applied on simulated loss data. Finally we will test the ability of presented methods to model loss data distributions.