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Combining sensometric and optometric tests and analyses
Králik, Roman ; Antoch, Jaromír (advisor) ; Pešta, Michal (referee)
In this research, we embarked on an in-depth exploration to discern the moderately damaged beer from its pristine counterparts by non-professional consumers in a controlled social setting. The approach adopted was underpinned by the application of rigorous sta- tistical testing methodologies, with an underlying potential for further refinement to yield enhanced insights. Crucially, the study identified that the outcomes could be skewed by phenomena such as taster fatigue and the saturation of ambient air with light-struck odours, thereby necessitating their consideration. It occurs even though tests were taken in spacious, well ventilated, room and each taster had enough, 3 meters at least, space around him. The research innovatively combines the optometric measurements of beer samples' absorption with the findings from structured tastings, enabling assumptions to be made about the absolute threshold of beer damage that a statistically significant number of tasters is able to reliably detect. For Pilsner Urquell, the threshold of percep- tible damage was delineated to be in excess of 0.067 absorption units (a.u.). For Excel- lent 11ř, the damage threshold was calculated to fall within the range of 46 to 67 thou- sandths of absorption units (a.u.). When we compared results divided by gender, we concluded...
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Superposition and thinning of counting processes in non-life insurance
Romaňák, Martin ; Pešta, Michal (advisor) ; Kříž, Pavel (referee)
The thesis examines a model for representing the number of claims after merging or splitting different lines of business of an insurance company. The model is based on count- ing processes, the Poisson and the renewal processes are considered in particular. The operations of superposition and thinning are the proposed solution to this problem. We present the well-known results that the Poisson processes are closed under superposition and several types of thinning and explore the necessary conditions for this statement to also hold for renewal processes. Specifically, the previous work on the superposition of renewal processes is studied and further clarified, and an original result is derived for two types of thinning of a renewal process. The theoretical results are then used to analyze real insurance data in a model situation when an insurance company wants to estimate the future number of claims after merging two of its lines of business. 1
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Two-dimensional distributions for given margins
Šťastný, Filip ; Pešta, Michal (advisor) ; Omelka, Marek (referee)
One of the tools for study of dependence between random variables are co- pulas. While modelling multidimensional variables it is possible using Sklar's theorem to model through copulas marginal distributions and relationship be- tween them separately, this approach thus enables us to split construction of multi-dimensional distributions into these two factors. With marginal distributi- ons fixed, the construction is consisting of appropriate copula choice only. This thesis deals with copulas in the case of two-dimensional distributions with conti- nuous fixed marginal distributions and is focused on parametrical copulas, mainly through Archimedean copulas. Basic properties of copulas with Sklar's theorem, which enables studying copulas in stochastic context, are presented here. Further, measures of dependence such as Kendall's tau, Spearman's rho and coeficients of tail dependence are in connection with copulas studied in this thesis. At the end, the thesis deals with methods of estimation unknown parameters, which are ilustrated on two examples. 1
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Bagging and regression trees in individual claims reserving
Janoušek, Jan ; Pešta, Michal (advisor) ; Mizera, Ivan (referee)
This diploma thesis focuses on the application of classification and regression trees, as well as bootstrap aggregating, to individual reserving in insurance. In the first part, we provide a summary of the theory and establish mathematical formalities that are sometimes overlooked in basic texts on these topics. We provide a comprehensive overview of the concepts, including a detailed discussion of their practical applications. In the second part, we build on existing research by extending the use of machine learning in individual claims reserving. Specifically, we expand on a prior article that only modeled the number of claims using classification trees. We also incorporate regression trees and bagging to model the size of each claim, resulting in more accurate reserve estimates. We achieve this by applying these techniques to insurance data and obtaining empirical distributions that allow us to calculate confidence intervals and quantiles. Ultimately, we determine the reserves needed for both the next year and the ultimate reserves. 1
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Chain-ladder method as maximum likelihood estimator in Poisson model
Wagner, Vojtěch ; Kříž, Pavel (advisor) ; Pešta, Michal (referee)
First, the distribution-free chain-ladder is introduced. Then, the Poisson model is in- troduced. It is proven that the total reserves for one accident year given by the maximum likelihood estimation applied to the Poisson model lead to the identical reserves as the reserves derived from the distribution-free chain-ladder used in the Poisson model. Later, inadequacies of the Poisson model are discussed. Hessian matrices of the log-likelihood evaluated at the Poisson estimators are analyzed. The question whether the inverse of the Fisher information matrix approximates the real covariance matrix of the Poisson esti- mators is explored. Comparing the variance of the total reserves derived from the inverse of the Fisher information and the real covariance matrix leads to negative conclusion, that the former does not approximate the latter well. 1
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Dependent zeros
Hanousek, Jan ; Pešta, Michal (advisor) ; Hendrych, Radek (referee)
This thesis investigates a specific type of non-negative time series containing a sig- nificant proportion of zeros. The goal of this work is to create a stochastic model which would be an appropriate representation of such time series. After examining existing theory about stochastic processes and the estimation of their parameters, we propose our own final models. Their suitability is tested using real-world data and the procedure shows that each model has its own advantages and limitations. Overall, the results are satisfactory, proving the credibility of the models and their applicability in practice and paving the way for possible further research on this topic. 1
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Operational risk and marked Poisson process
Váchová, Karla ; Pešta, Michal (advisor) ; Dvořák, Jiří (referee)
The subject of this bachelor thesis entitled "Operational risk and marked Poisson process" is the modelling of operational risk using marked Poisson process. The Poisson process is a type of a point process that models randomly distributed points on some underlying space. Because of its mathematical properties, it is a quite frequently used model in biology, astronomy, ecology or economics, for example. This bachelor thesis describes its basic properties and uses the marked Poisson process to model loss frequency and severity belonging to bank's operational risk. 1
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Truncated random vectors
Raab, Petr ; Pešta, Michal (advisor) ; Komárek, Arnošt (referee)
This bachelor thesis deals with truncated random vectors, distributions and properties of theirs. Truncated random vectors theory is then used to solve problem of delayed reporting of non-life insurance claims. At the of this thesis there are shown properties and behaviour of the estimators, which are constructed in this thesis, while being applicated on real life data from vehicle accident insurancy. 1
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Copulae for non-continuous distributions
Mifkovič, Matej ; Pešta, Michal (advisor) ; Omelka, Marek (referee)
Copulas are a popular choice when assessing the dependence structure between continuous random variables. However, major difficulties arise as soon as one of the random variables is non-continuous. This thesis introduces the basics of copula theory based on the cited literature. The main focus of this thesis is to introduce the reader to the field of non- continuous copula modelling and highlight all major issues. At the same time, empirical evidence with discussion is presented to suggest that copula modelling and inference may be a viable option when additional care and caution are applied. Afterwards, accumulated theoretical knowledge is demonstrated on real-world data concerning bike-sharing.
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