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Copula-based multivariate association measures and tail coefficients
Kika, Vojtěch ; Omelka, Marek (advisor) ; Veraverbeke, Noel (referee) ; Fuchs, Sebastian (referee)
The dependence structure of a d-variate random vector X is a very complex notion which is fully described by the distribution of the random vector. Alternatively, it suffices to look into the corresponding copula function of X, as it ignores the marginal distributions of X but still fully describes the dependence structure. However, a copula is a function defined on the d-dimensional hypercube [0, 1]d with values in the interval [0, 1]. As such, it might be too complex for practical use and one would prefer to have tools that can translate the information from the copula function into a simpler indicator. In particular, of interest might be an association measure, that is, a single number that describes the tendency of the components of X to simultaneously take large or small values. Coefficients like Kendall's tau or Spearman's rho, used to measure (strength of) an association between two random variables, were thoroughly studied and described in the middle of 20th century. Requirements on bivariate association measures are well-known. However, generalization of such measures into higher dimensions is not very straightforward and brings discussion on the desirable properties. In addition, bivariate association measures can be often generalized in multiple manners. The same holds true if one wants to...
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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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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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