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Generalized Linear Models in Reserving Risk
Zboňáková, Lenka ; Pešta, Michal (advisor) ; Branda, Martin (referee)
In the presented thesis we deal with the generalized linear models framework in a claims reserving problem. Claims reserving in non-life insurance is firstly described and the considered class of models is introduced. Consequently, this branch of stochastic modelling is implemented in the reserving setup. For computation of the risk associated with claims reserving, we need a predictive distribution of future liabilities in order to evaluate risk measures such as Va- lue at Risk and Conditional Value at Risk. Since datasets in non-life insurance commonly consist of a small number of observations and estimation of predictive distributions can be complicated, we adopt a bootstrap method for this purpose. Model fitting, simulations and consequent measuring of the reserving risk are performed within the use of real-life data. Based on this, an analysis of fitted models and their comparison together with graphical outputs is included. 1
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Bayesian Approaches to Stochastic Reserving
Novotová, Simona ; Pešta, Michal (advisor) ; Branda, Martin (referee)
In the master thesis the issue of bayesian approach to stochastic reserving is solved. Reserving problem is very discussed in insurance industry. The text introduces the basic actuarial notation and terminology and explains the bayesian inference in statistics and estimation. The main part of the thesis is framed by the description of the particular bayesian models. It is focused on the derivation of estimators for the reserves and ultimate claims. The aim of the thesis is to show the practical uses of the models and the relations between them. For this purpose the methods are applied on a real data set. Obtained results are summarized in tables and the comparison of the methods is provided. Finally the impact of a prior distribution on the resulting reserves is showed. Powered by TCPDF (www.tcpdf.org)
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Expectile regression
Ondřej, Josef ; Komárek, Arnošt (advisor) ; Pešta, Michal (referee)
In this thesis we present an alternative to quantiles, which is known as expectiles. At first we define the notion of expectile of a distribution of ran- dom variable and then we show some of its basic properties such as linearity or monotonic behavior of τ-th expectile eτ in τ. Let (Y, X), Y ∈ R, X ∈ Rp be a ran- dom vector. We define conditional expectile of Y given X = x, which we denote eτ (Y |X = x). We introduce model of expectile regression eτ (Y |X = x) = x⊤ βτ , where βτ ∈ Rp and we examine asymptotic behavior of estimate of the regression coefficients βτ and ways how to calculate it. Further we introduce semiparametric expectile regression, which generalizes the previous case and adds restrictions on the estimate of the regression coefficients which enforce desired properties such as smoothness of fitted curves. We illustrate the use of theoretical results on me- chanographic data, which describe dependence of power and force of a jump on age of children and adolescents aged between 6 and 18. Keywords: expectiles, expectile regression, quantiles, penalized B-splines 1
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Computing the aggregate claims distribution
Dlouhá, Veronika ; Pešta, Michal (advisor) ; Cipra, Tomáš (referee)
In this bachelor thesis I study the distribution of aggregate claims. First I introduce the topic and present main models. Then, I show some distributions used for modeling number and amound of claims with estimates of their parameters. Next I get to compound distribution and its basic charakteristics. In other parts of the thesis I study the probability of aggregate claims using Panjer recursion and fast Fourier transform and apply the thoery to examples. Finally I mention some methods to approximate aggregate claims using known distribution.
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Multidimensional Credibility
Zhuravova, Nadezda ; Mazurová, Lucie (advisor) ; Pešta, Michal (referee)
The aim of this graduation work is theoretically describe and also demonstrate the practical application of the theory of credibility in the multidimensional case. This theory is one of the most frequently used methods for calculating premiums, expected claims frequency or the expected average amount of damage. In this work we describe multidimensional Bühlmann-Straub credibility model and two- dimensional model with a known distribution. For each of these models we derive credibility estimate and examples of using these estimates in practice. Both models are compared on simulated data. Powered by TCPDF (www.tcpdf.org)
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Least Squares Alternatives
Gerthofer, Michal ; Pešta, Michal (advisor) ; Kulich, Michal (referee)
In the present thesis we deal with the linear regression models based on least squares. These methods are discussed in two groups. The first one focuses on three primary aproaches devided by occurrence of errors in variables. The traditional approach penalizes only the misfit in the de- pendent variable part and is called the ordinary least squares (OLS). An opposite case to the OLS is represented by the data least squares (DLS), which allow corrections only in the explanatory variables. Consecutively, we concentrate ourselves on the total least squares approach (TLS) mi- nimizing the squares of errors in the values of both dependent and independent variables. Finally, we give attention to next group of methods whit high breakdown point, which deal with signifi- cance of the individual observations (least weighted squares) and elimination of outlying obser- vations (least trimmed squares). The main purpose of this work is to describe and compare these models, their assumptions, characteristics, properties of estimates and show them on real data. 1
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Zobecněné lineární a aditivní modely v pojišťovnictví
Rusnák, Peter ; Branda, Martin (advisor) ; Pešta, Michal (referee)
In this thesis we describe the theory of generalized linear models and demon- strate its applications in non-life insurance. We also introduce some methods com- monly used to estimation of regression parameters and hypothesis testing . Further- more, we discuss possible extensions of GLM by introducing tools for reparametriza- tion of predictors which leads to new classes of models, concretely to segmented generalized linear models and generalized additive models. Consequently, we derive models appropriate for actuarial praxis using the real insurance data. In practical part of this thesis we illustrate the use of appropriate software for calculating the parameters of GAM and find way how to use open source statistical program .
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Generalized Linear Models in Insurance
Staněk, Petr ; Mazurová, Lucie (advisor) ; Pešta, Michal (referee)
In the present thesis we study Generalized linear mo- dels with their application in the insurance. Our specifiation is in using Poisson and Binomial distribution. Our goal is application of the theoretical knowledge in the vehicle insurace, claim reserving and survival model. Keywords: Link, linear predictor, response variable, explanatory variable, likelihood function, quasilikelihood function, deviance, re- ziduals, survival model, run-off triangle, canonical link, Poisson dis- tribution, binomial distribution, dispersion parameter, variance func- tion. 1
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Claims count modeling in insurance
Škoda, Štěpán ; Branda, Martin (advisor) ; Pešta, Michal (referee)
1 Abstract: The present work investigates techniques of insurence ratemaking accor- ding to the claims counts of policyholders on the basis of information contained in policies. At the beginning, we provide a closer examination of the theory of genera- lized linear models, which have wide range of applications in the field of actuarial modeling. The second chapter presents the basic Poisson regression model as well as some particular verification methods. Specifically, deviance and Wald test could be found here and furthermore also important results for residuals. The third chapter contains information on alternative approaches to modeling the claim frequencies and at the end the GEE method, that can be applied in case of panel data, is de- scribed. The numerical study based on real insurace data in last part of this diploma thesis illustrate's previously described techniques which were obtained with the help of statistical software SAS.
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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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