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Dense zeros
Hanousek, Jan ; Pešta, Michal (advisor) ; Cipra, Tomáš (referee)
This research focuses on a special type of time series data where a significant propor- tion of values is zero. The aim is to develop a statistical model that accurately captures the behavior of such data. By exploring existing theories on GARCH and MEM models, new models together with derivation of important theoretical properties are proposed. To assess their effectiveness, they are tested on real-world data. This evaluation reveals that each model has its own strengths and weaknesses. The overall results are promis- ing, proving the models' validity and real-world applicability, opening doors for further exploration in this area. 1
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Universality and consistency theorems for neural networks
Raab, Petr ; Pešta, Michal (advisor) ; Hlubinka, Daniel (referee)
The work deals with neural networks and deep learning models, where the author attempts to view neural networks as a statistical model similar to generalized linear models. After introducing this model and introducing the notation, the work focuses on the ability of neural networks to approximate continuous functions, with a proof of the universality theorem presented. Subsequently, the asymptotic properties of neural networks are examined, and using network estimation, their consistency and asymptotic normality are also proven. These two properties are precisely the subject of investigation in a simulation study on generated data. 1
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Multiplicative Error Models
Krahulík, Matyáš ; Pešta, Michal (advisor) ; Hudecová, Šárka (referee)
This thesis is devoted to the so-called multiplicative error models (MEM), which are used to model non-negative time series, most often in the financial sector. The first chapter focuses on ARCH and GARCH models, which do not belong to the group of multiplicative error models, but are closely related to them. The second chapter focuses directly on the MEM and their further extensions, such as zero-augmented MEM (ZA-MEM) or semiparametric MEM (SpMEM). These models are first defined and then methods for parameter estimation in these models are presented. In the third chapter, which contains the practical part of the thesis, the practices from the second chapter are applied to real data in the form of a time series of claims from one of the Czech insurance companies. In the conclusion, further extensions to the the applications of the MEM to insurance or other data are proposed. 1
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Wild binary segmentation
Lasota, Jakub ; Pešta, Michal (advisor) ; Maciak, Matúš (referee)
The goal of this work is to describe some (multiple) change-point detection methods that aim to estimate the total number and locations of structural changes in the data. From the variety of all change-point detection methods, only binary segmentation and wild binary segmentation are explained. To enhance the understanding, the work contains a few illustrative examples that try to show the strengths and weaknesses of each method. The practical part of the work focuses on using and comparing both methods with various parameter choices on daily logarithmic returns of the Zoom Video Communications stock. 1
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Homoscedasticity tests in one-way classification
Gulík, Matyáš ; Komárek, Arnošt (advisor) ; Pešta, Michal (referee)
The bachelor thesis deals with tests of variance equality in the context of simple classification. It focuses on three tests commonly used in practice. The thesis first provides an overview of concepts and knowledge from probability theory, which are utilized in subsequent chapters. Additionally, a one-way analysis of variance is introduced, which is crucial for the tests of variance equality. In the main part of the thesis, the Levene test is derived, followed by the Brown-Forsythe test, which is its modification. The Bart- lett test is also presented. Finally, simulations were conducted using the R program to determine the ability of the tests to maintain the desired significance level. 1
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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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