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Hawkes process
Feketeová, Adela ; Pešta, Michal (advisor) ; Flimmel, Daniela (referee)
This bachelor thesis is dedicated to Hawkes process. It is divided into six chapters. Chapter one consists of an introduction into the theory of random processes and a de- scription of Poisson process, the second chapter is dedicated to the definition of Hawkes process and its properties. The third chapter contains simmulation methods of the Haw- kes process. Chapter four further describes the Hawkes process parameter estimation and the fifth chapter consists of model goodness of fit testing for observed data. Lastly, the sixth chapter contains practical example of fitting the Hawkes model for real data and a descrition of R package hawkesbow used. 1
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Truncated counting processes
Přítel, Ondřej ; Pešta, Michal (advisor) ; Prokešová, Michaela (referee)
The aim of this thesis is the prediction of insurance events under the condition that the data related to the occurrence of the events is truncated. The nature of the truncation lies in the fact that in the present we observe only those events that were already reported to the insurance company. Occurrences and reporting are modeled by a two-dimensional non-homogeneous Poisson process. The intensity of occurrences is derived from Kingman's Displacement theorem and is computed as a convolution of the intensity of reporting and the density of the delay in between occurrences and reporting. The estimations of the parametric function of the intensity of reporting and the distribution are preformed using the maximum likelihood method. In addition, theoretical background concerning counting processes primarily directed to the Poison processes is discussed in this thesis. 1
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Correlation analysis and betting odds
Josefus, Pavel ; Pešta, Michal (advisor) ; Večeř, Jan (referee)
This bachelor's thesis focuses on a statistical method called correlation analysis. The aim of the thesis is to explain various correlation coefficients such as Pearson's correlation coefficient, point biserial correlation, Spearman's rank correlation coefficient and Kendall's rank order correlation coefficient. The thesis presents confidence intervals for each of them and also tests hypotheses about correlation coefficients. The practical part of the thesis applies established methods to real data concerning courses on women's tennis match results. 1
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Dynamic prediction in survival analysis
Mečiarová, Kristína ; Komárek, Arnošt (advisor) ; Pešta, Michal (referee)
Often the motivation behind building a statistical model is to provide prediction for an outcome of interest. In the context of survival analysis it is important to distingu- ish between two types of time-varying covariates and take into careful consideration the appropriate type of analysis. Joint model for longitudinal and time-to-event data, in con- trast to standard Cox model, enables to account for continuous change of the covariate over time in the survival model. In this thesis two examples of joint models are presen- ted, the shared random-effect model and the joint latent class model. Bayesian estimation of the model parameters and summary of methodology for dynamic prediction of indi- vidual survival probability is provided for the first one of the aforementioned types of models. Application of the theoretical knowledge is illustrated in the analysis of the data on primary biliary cirrhosis. The impact of number of patients, number of longitudinal measurements and per-cent of censoring on the quality of prediction and estimates of the model parameters is examined in the simulation study. 1
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Truncated marked processes
Hrbáčová, Daniela ; Pešta, Michal (advisor) ; Dvořák, Jiří (referee)
This thesis explores the use of marked stochastic processes in the context of delayed reporting of claims in non-life insurance. The focus is on estimating the intensity of the claim occurrence process using the ν-transform of the claim reporting process. The first part provides the theoretical background, including the introduction of the Poisson process and the concept of marking. The ν-transform is defined and a special case of the ν-transform is applied in an example. As well as there is presented an approach how to handle with truncated data. The second chapter applies these theoretical concepts to real-world data from Motor Third Party Liability insurance. The result is a formula for estimating the intensity of the occurrence process based on the estimated intensity of the claim reporting process and the estimated truncated conditional density of delays given reporting times. While the approach is computationally intense, it has practical applica- tions in estimating claim reserves for insurance companies. Future work could expand on this approach by considering more complex cases, such as time-varying conditional dis- tribution of delays or including on input nonhomogeneous Poisson, or even more complex processes. Finishing the claim reserve calculation would be also beneficial. Overall, this thesis...
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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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