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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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Reserving in Non-Life Insurance
Těšínský, Jiří ; Gerthofer, Michal (advisor) ; Zimmermann, Pavel (referee)
This thesis deals with reserving issues in non-life insurance. The main aim is to compare methods used to estimate outstanding claims reserves and delineate their advantages, disadvantages, extensions and limitations. Initially, in the theoretical section, there are mathematically formulated chosen models and their assumptions, which are then applied to real data in the practical section. Finally, the findings acquired from both parts are utilized for the comparison of applicability of methods and the identification of their further distinctions and commonalities.
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Extreme value theory
Pelinka, Adam ; Čabla, Adam (advisor) ; Gerthofer, Michal (referee)
Extreme value theory is a modern statistical method for modelling events with a very low probability. During the analysis, we deal with convergence of distribution of these extremal events to their limit distributions. These distributions are generalized extreme value distribution and generalized Pareto distribution, which estimate tails of empirical probability distribution, where extremal events occur. In the last years, extreme value theory is used in many fields of study, e.g. in estimating financial risk or in estimating size of floods. In this work, two methods of modelling extremal events are presented - block maxima method and peaks over threshold method. Both methods are used during the real data analysis of daily discharge of river Vltava and estimated models are summarized. Although both used methods give slightly different results, choice of the appropriate model is not clear.
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Claims reserving within the panel data framework
Gerthofer, Michal ; Pešta, Michal (advisor) ; Cipra, Tomáš (referee)
In the presented thesis the issue of dependency between response variables within the subjects in the generalized linear models framework is investigated. Reserving in non-life insurance is a key factor for the financial position of a company. The text introduces the basic actuarial notation, terminology and methods. The main part is focused on panel data framework, especially Generalized Linear Mixed Models (GLMM) as well as Generalized Estimating Equations (GEE), and their application on claims reserving. The aim of this thesis is to show the advantages, disadvantages, limitations and the comparison of these approaches on representative datasets, which were chosen according to results obtained from whole database analysis. Significant focus is on model selection and diagnostics used for this purpose. Finally, the obtained results are summarized in tables, figures and the comparison of the methods is provided. 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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