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Extreme Value Distribution Parameter Estimation and its Application
Holešovský, Jan ; Picek,, Jan (referee) ; Antoch,, Jaromír (referee) ; Michálek, Jaroslav (advisor)
The thesis is focused on extreme value theory and its applications. Initially, extreme value distribution is introduced and its properties are discussed. At this basis are described two models mostly used for an extreme value analysis, i.e. the block maxima model and the Pareto-distribution threshold model. The first one takes advantage in its robustness, however recently the threshold model is mostly preferred. Although the threshold choice strongly affects estimation quality of the model, an optimal threshold selection still belongs to unsolved issues of this approach. Therefore, the thesis is focused on techniques for proper threshold identification, mainly on adaptive methods suitable for the use in practice. For this purpose a simulation study was performed and acquired knowledge was applied for analysis of precipitation records from South-Moravian region. Further on, the thesis also deals with extreme value estimation within a stationary series framework. Usually, an observed time series needs to be separated to obtain approximately independent observations. The use of the advanced theory for stationary series allows to avoid the entire separation procedure. In this context the commonly applied separation techniques turn out to be quite inappropriate in most cases and the estimates based on theory of stationary series are obtained with better precision.
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Analysis of extreme values
Vyhlídka, Jan ; Hendrych, Radek (advisor) ; Antoch, Jaromír (referee)
The goal of this thesis is to introduce basic concepts of the extreme value theory. The first chapter describes two fundamentally different approaches - block maxima and peaks over threshold models. Furthermore, it presents generalized extreme value distribution and generalized Pareto distribution. Moreover, relevant theorems and characteristics that are tied to these probabilistic distributions are discussed. The second chapter is a survey of various methods of parameter estimation of discussed distributions. The last chapter shows a simple application of how extreme value theory can be applied in finance on selected shares listed on the Prague Stock Exchange.
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Methods of modelling and statistical analysis of an extremal value process
Jelenová, Klára ; Volf, Petr (advisor) ; Branda, Martin (referee)
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.
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Extreme Value Theory in Actuarial Sciences
Jamáriková, Zuzana ; Mazurová, Lucie (advisor) ; Antoch, Jaromír (referee)
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.
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Modelování velkých škod
Petrová, Barbora ; Pešta, Michal (advisor)
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
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Modelování velkých škod
Petrová, Barbora ; Pešta, Michal (advisor)
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
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Operational risk modelling
Mináriková, Eva ; Mazurová, Lucie (advisor) ; Hlubinka, Daniel (referee)
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.
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Modelování velkých škod
Zuzáková, Barbora ; Pešta, Michal (advisor) ; Hlubinka, Daniel (referee)
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
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Extreme Value Theory in Actuarial Sciences
Jamáriková, Zuzana ; Mazurová, Lucie (advisor) ; Antoch, Jaromír (referee)
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.
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Methods of modelling and statistical analysis of an extremal value process
Jelenová, Klára ; Volf, Petr (advisor) ; Branda, Martin (referee)
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.
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