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Beta regression
Štěpán, Marek ; Hudecová, Šárka (advisor) ; Omelka, Marek (referee)
The thesis deals with a beta regression model suitable for analysing data whose range of values is the interval (0, 1). The model assumes a conditional beta distribution for the response given covariates, and its structure is similar to generalised linear models. The model is defined and its basic properties are investigated. The asymptotic distribution of the maximum likelihood estimates is provided. A possible extension to situations where the response in the data attains one of the boundary values is considered and referred to as c-inflated beta regression model. For both models, statistical inference and model diagnostics are discussed. The practical part of the thesis involves two Monte Carlo studies and two real data analyses. The first simulation study compares the performance of the global goodness-of-fit measures for link selection, while the second study explores various approaches to the analysis of the inflated beta distribution response. Alternative initial values are proposed for the cases where the algorithm did not converge. The practical usage of the model is illustrated on a model of proportions of tertiary educated people in European countries, and the proportion of household income spent on education in the Philippines. 1
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The Kelly Criterion
Kálosi, Szilárd ; Omelka, Marek (advisor) ; Hlávka, Zdeněk (referee)
The present work is devoted to the Kelly criterion, which is a simple method for choosing the amount of the bet for gambles with a positive expected value. In the first part of the work we introduce the mathematical explanation of the criterion, examine the capital after $n$ trials as a function of the bet, the long-run rate of return and asymptotical properties of the capital growth. In the second part we attempt to generalize the Kelly criterion from the first part for some other situations. Examples for a simple game and generalized situations illustrating the properties of the Kelly criterion and results from previous parts compose the last part of the work.
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Paradoxes in Probability Theory and Mathematical Statistics
Klouparová, Zdeňka ; Omelka, Marek (advisor) ; Stibůrek, David (referee)
In this work I deal with few selected paradoxes related to games. First of all I explain a paradox directly from the game theory field. I show that kids' game about matching fingers is advantageous for one player although it seems to be fair for both players at the first sight. Second example touches war troubles with hidden objects. In the second chapter I explain the Gladiator paradox and I try to find the best order in which the gladiators should be sent to an arena to fight. Finally, I also touch the paradox of transitivity and explain how the game with nontransitive dices works. Keywords: Game theory paradox, Gladiator paradox, paradox of transitivity 1
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The Depth of Functional Data.
Nagy, Stanislav ; Hlubinka, Daniel (advisor) ; Omelka, Marek (referee)
The depth function (functional) is a modern nonparametric statistical analysis tool for (finite-dimensional) data with lots of practical applications. In the present work we focus on the possibilities of the extension of the depth concept onto a functional data case. In the case of finite-dimensional functional data the isomorphism between the functional space and the finite-dimensional Euclidean space will be utilized in order to introduce the induced functional data depths. A theorem about induced depths' properties will be proven and on several examples the possibilities and restraints of it's practical applications will be shown. Moreover, we describe and demonstrate the advantages and disadvantages of the established depth functionals used in the literature (Fraiman-Muniz depths and band depths). In order to facilitate the outcoming drawbacks of known depths, we propose new, K-band depth based on the inference extension from continuous to smooth functions. Several important properties of the K-band depth will be derived. On a final supervised classification simulation study the reasonability of practical use of the new approach will be shown. As a conclusion, the computational complexity of all presented depth functionals will be compared.
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Small sample asymptotics
Tomasy, Tomáš ; Sabolová, Radka (advisor) ; Omelka, Marek (referee)
In this thesis we study the small sample asymptotics. We introduce the saddlepoint approximation which is important to approximate the density of estimator there. To derive this method we need some basic knowledge from probability and statistics, for example the central limit theorem and the M- estimators. They are presented in the first chapter. In practical part of this work we apply the theoretical background on the given M-estimators and selected distribution. We also apply the central limit theorem on our estimators and compare it with small sample asymptotics. At the end we show and summarize the calculated results.
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Kelly criterion in portfolio selection problems
Dorová, Bianka ; Kopa, Miloš (advisor) ; Omelka, Marek (referee)
In the present work we study portfolio optimization problems. Introduction is followed by chapter 2, where we introduce the concept of utility function and its relationship to the investor's risk attitude. To solve the optimization problem we consider the Markowitz portfolio optimization model and the Kelly criterion, which are recalled in the fourth and fifth chapter. The work also contains an extensive numerical study. Using the optimization software GAMS we solve portfolio optimization problems. We consider a portfolio problem with (and without) allowed short sales. We compare the obtained portfolios and we discuss whether Kelly optimal portfolio is a special case of the Markowitz optimal portfolio for the special value of the minimum expected return.
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Statistical applications of urn models
Navrátil, Radim ; Pawlas, Zbyněk (advisor) ; Omelka, Marek (referee)
This work shows various applications of urn models in practice. First, basic properties of the occupancy distribution are derived together with its asymptotic approximation. This model is applied and generalized in the theory of database systems for records search from a given database. An application to random texts is mentioned, namely the computation of the expected number of missing and common words in random texts. There are presented exact formulas, their asymptotic approximations and the approximations via occupancy distribution. Then, some urn models, which are used in the randomized response theory for finding out respondents' answers to sensitive questions, are described. These models are compared according to their accuracy and respondents' goodwill to answer. Finally, two non-parametric tests of empty boxes are derived, one for the hypothesis whether a random sample comes from a given population and the second for the hypothesis whether two independent random samples come from the same population. The powers of these tests are compared with commonly used tests for these hypotheses.
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Methods of artificial intelligence and their use in prediction
Šerý, Lubomír ; Omelka, Marek (advisor) ; Krtek, Jiří (referee)
Title: Methods of artificial intelligence and their use in prediction Author: Lubomír Šerý Department: Department of Probability and Mathematical Statistics Supervisor: Ing. Marek Omelka, Ph.D., Department of Probability and Mathe- matical Statistics Abstract: In the presented thesis we study field of artificial intelligence, in par- ticular we study part dedicated to artificial neural networks. At the beginning, concept of artificial neural networks is introduced and compared to it's biological base. Afterwards, we also compare neural networks to some generalized linear models. One of the main problems of neural networks is their learning. Therefore biggest part of this work is dedicated to learning algorithms, especially to pa- rameter estimation and specific computational aspects. In this part we attempt to bring in an overview of internal structure of neural network and to propose enhancement of learning algorithm. There are lots of techniques for enhancing and enriching basic model of neural networks. Some of these improvements are, together with genetic algorithms, introduced at the end of this work. At the very end of this work simulations are presented, where we attempt to verify some of the introduced theoretical assumptions and conclusions. Main simulation is an application of concept of neural...
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