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Generalized Stable Models in Finance
Chovanec, Róbert ; Štěpán, Josef (referee) ; Klebanov, Lev (advisor)
In this contribution, a basic theoretical approach to stable laws is described. There are mentioned some definitions of the stable distributions, properties and behavior of stable distributed random variables. Next, conditional modeling under the stable laws are analyzed. One can find homoskedastic (ARMA) and heteroskedastic (GARCH) structures. The GARCH models are explained partly for the Gaussian case too. An empirical application of this paper is based on comparison between the models, established in theoretical part, under the normal, and stable distribution respectively, built on real data from energetics. One issues from unconditional, then continues with conditional ARMA and finally, there are mixed ARMA-GARCH models. The results of interpreted statistical analysis demonstrate that the models based on the stable distribution matched the empirical distribution better than the the models based on the Gaussian distribution.
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Statistical methods for determination of the weight of evidence in the identification process
Slovák, Dalibor ; Klebanov, Lev (referee) ; Zvárová, Jana (advisor)
In the present work we study an identification of culprit and assesment of evidence against him. At first we define a simple model called the island problem and we derive the weight-of-evidence formula in its basic form. In the next chapters we analyse several modifications of island problem and related issues. We find how we can deal with uncertainty about basic parametres of model, like size of population. We investigate possibility of inclusion of different errors or influence of relatedness and subpopulation structure into model. At the close we enlarge mixtures of DNA, including deriving and programming of appropriate formulas.
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On measures of biodiversity and their applications
Horáček, Martin ; Klebanov, Lev (referee) ; Zvárová, Jana (advisor)
In the present work we study problems that are connected to measures of biodiversity. Our approach to these problems is based on theory of information, namely on so called generalized entropies, also known as f-entropies. If we add a restriction on these f-entropies, we receive a set of functions that we call f-diversities, which satisfy usual requirements for functions that are intended to measure biodiversity. Moreover, we show that many measures of diversity that are used in practice are either f-diversities or continuous functions of some f-diversity. We study various measures of diversity and we investigate whether they can or can not be included in this framework. We introduce a concept that can be used to evaluate sensitivity to changes of various measures of diversity. This concept might be useful to study behavior of different measures of biodiversity. We also study issues connected to estimates of biodiversity measures and we investigate some types of estimates. For one type we present a slightly modified approach that can result in better estimates for some measures. We focus on theoretical investigation of measures of diversity, however most of our results are practically applicable.
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Stochastic Models with Sums Containing a Random Number of Random Variables
Vališ, Pavel ; Dostál, Petr (referee) ; Klebanov, Lev (advisor)
The thesis studies stochastic models with sums of a random number of random variables. It shows examples of such models including utilization in economics. It describes stable and v-stable distributions with conditions of their existence and with some their properties. Analogies of these distributions are introduced for discrete random variables and some properties and examples are given. Possible series representations of such distributions are mentioned. Analogies of the Law of Large Numbers and the Central Limit Theorem for a random number of random variables are also described.
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