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Multidimensional statistics and applications to study genes
Bubelíny, Peter ; Klebanov, Lev (advisor) ; Jurečková, Jana (referee) ; Kalina, Jan (referee)
Title: Multidimensional statistics and applications to study genes Author: Mgr. Peter Bubelíny Department: Department of probability and mathematical statistics Supervisor: prof. Lev Klebanov, DrSc., KPMS MFF UK Abstract: Microarray data of gene expressions consist of thousands of genes and just some tens of observations. Moreover, genes are highly correlated between themselves and contain systematic errors. Hence the magnitude of these data does not afford us to estimate their correlation structure. In many statistical problems with microarray data, we have to test some thousands of hypotheses simultaneously. Due to dependence between genes, p-values of these hypotheses are dependent as well. In this work, we compared conve- nient multiple testing procedures reasonable for dependent hypotheses. The common manner to make microarray data more uncorrelated and partially eliminate systematic errors is normalizing them. We proposed some new normalizations and studied how different normalizations influence hypothe- ses testing. Moreover, we compared tests for finding differentially expressed genes or gene sets and identified some interesting properties of some tests such as bias of two-sample Kolmogorov-Smirnov test and interesting behav- ior of Hotelling's test for dependent components of observations. In the end of...
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Generalized stable distributions and their applications
Slámová, Lenka ; Klebanov, Lev (advisor) ; Maslowski, Bohdan (referee) ; Korolev, Victor (referee)
Title: Generalized stable distributions and their applications Author: Mgr. Lenka Slámová, MSc. Department: Department of probability and mathematical statistics Supervisor: Prof. Lev Klebanov, DrSc. Abstract: This thesis deals with different generalizations of the strict stability property with a particular focus on discrete distributions possessing some form of stability property. Three possible definitions of discrete stability are introduced, followed by a study of some particular cases of discrete stable distributions and their properties. The random normalization used in the definition of discrete stability is applicable for continuous random variables as well. A new concept of casual stability is introduced by replacing classical normalization in the definition of stability by random normalization. Examples of casual stable distributions, both discrete and continuous, are given. Discrete stable distributions can be applied in discrete models that exhibit heavy tails. Applications of discrete stable distributions on rating of scientific work and financial time series modelling are presented. A method of parameter estimation for discrete stable family is also introduced. Keywords: discrete stable distribution, casual stability, discrete approximation of stable distribution
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Heavy tailed distributions and their applications to finance
Korbel, Michal ; Klebanov, Lev (advisor) ; Janák, Josef (referee)
In this work we describe heavy tailed distributions. We show conditions necessary and sufficient for their existence. First we study the product of random number of random variables and their convergence to the Pareto distribution. We also show graphs that concur this theorem. Next we define stable distributions and we study their usefulness for approximating of sum of random number of random variables. We also define Gauss and infinitely divisible random variables and we show conditions for their existence. We also show that the only geometric stable distribution following the stable law are strictly geometric stable or improper geometric stable distributions. In the end we study applications of stable distributions in finance and we show example for their usage in computing VaR. Powered by TCPDF (www.tcpdf.org)
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Bivariate distributions
Bednárik, Vojtěch ; Pawlas, Zbyněk (advisor) ; Klebanov, Lev (referee)
The thesis deals with three selected constructions of bivariate distributions. The first approach is to use the Fréchet bounds, which determine restrictions on the distribution function and the correlation coefficient of bivariate distribution. The second construction is the Plackett distribution which is a class of distributions containing the Fréchet bounds and the member corresponding to independent random variables. The third construction is a trivariate reduction method that is used for a construction of bivariate gamma, exponen- tial and Poisson distribution. Only bivariate Dirichlet distribution has slightly different construction. For the last four mentioned distributions the following basic characteris- tics are derived: density function, marginal distributions, correlation coefficient and some conditional moments, in case of exponential and Dirichlet distribution even conditional distribution. 1
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Options under Stable Laws
Karlová, Andrea ; Volf, Petr (advisor) ; Klebanov, Lev (referee) ; Witzany, Jiří (referee)
Title: Options under Stable Laws. Author: Andrea Karlová Department: Department of Probability and Mathematical Statistics Supervisor: Doc. Petr Volf, CSc. Abstract: Stable laws play a central role in the convergence problems of sums of independent random variables. In general, densities of stable laws are represented by special functions, and expressions via elementary functions are known only for a very few special cases. The convenient tool for investigating the properties of stable laws is provided by integral transformations. In particular, the Fourier transform and Mellin transform are greatly useful methods. We first discuss the Fourier transform and we give overview on the known results. Next we consider the Mellin transform and its applicability on the problem of the product of two independent random variables. We establish the density of the product of two independent stable random variables, discuss the properties of this product den- sity and give its representation in terms of power series and Fox's H-functions. The fourth chapter of this thesis is focused on the application of stable laws into option pricing. In particular, we generalize the model introduced by Louise Bachelier into stable laws. We establish the option pricing formulas under this model, which we refer to as the Lévy Flight...
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Normality and its testing
Hájek, Štěpán ; Bašta, Milan (advisor) ; Klebanov, Lev (referee)
This thesis is concerned with normality and its testing. We often encounter with this topic when using statistical tests and models. Among others, examples such as t tests, analysis of variance and linear regression might be given. In this thesis these tests and models are overviewed and the consequences of the violation of the normality assumption are briefly mentioned. The following section describes statistical tests of normality. For example Shapiro-Wilk test or Anderson-Darling test are explored. For each test of normality is given test statistic and conditions for rejection of the null hypothesis. The last section provides a simulation study. The first part of this study is devoted to exploring whether the empirical relative frequency of Type I error corresponds to the nominal significance level of the test. The second part of the simulation study explores the power of normality tests against various alternatives. The results are summarized and discussed. 1
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Introduction to Order Statistics Theory
Hanuš, Antonín ; Kulich, Michal (advisor) ; Klebanov, Lev (referee)
This thesis deals with the theory of order statistics. Its aim is to summarize the basic knowledge concerning the distribution of the order statistics of random variables that are absolutely continuous with respect to the Lebesgue Measure and afterwards use those order statistics for some specific distributions. The first chapter describes the derivation of the density and distribution function of order statistics in several ways as well as dealing with some functions of order statistics and their conditional distribution. The second chapter is devoted to the moments of order statistics and formulae for their calculation and to the relations between them. In the conclusion the previous theoretical findings are applied to the uniform, exponential and normal distributions. 1
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Deterministické a stochastické modely v molekulární a buněčné biologii
Krasnovský, Pavol ; Vejchodský, Tomáš (advisor) ; Klebanov, Lev (referee)
This thesis presents the main methods that are used to model the time evolution of the number of molecules in a cell. Two of the main aims in cell biology are to compute first the transi- tion probability function and second the density of the invariant measure. These two problems imply a number of conditions and hence we also include the ergodic theory and theory of the invariant measure. We use two illustrative examples of the application of the previously mentioned theories. We verify the necessary and sufficient conditions for the computation of the transition probability function and the density of the invariant measure in case of two types of a chemical system. The probability function and the density are then given by a numerical solution to the Fokker-Planck equation in both the dynamic and the stationary case. Furthermore, we compare the obtained solu- tions to the results from the Monte Carlo simulation. We find that the solutions give almost identical results as the Monte Carlo simulation. At the end of this thesis, we formulate and analyze a chemical system represented by a human cell infected by an influenza virus. Given the complexity of the sys- tem, we compute the results using the Monte Carlo method. In addition we define this problem by a stochastic differential equation with random...
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Large deviations and their applications in insurance mathematics
Fuchsová, Lucia ; Pawlas, Zbyněk (advisor) ; Klebanov, Lev (referee)
Title: Large deviations and their applications in insurance mathematics Author: Lucia Fuchsová Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Zbyněk Pawlas, Ph.D. Supervisor's e-mail address: Zbynek.Pawlas@mff.cuni.cz Abstract: In the present work we study large deviations theory. We discuss heavy-tailed distributions, which describe the probability of large claim oc- curence. We are interested in the use of large deviations theory in insurance. We simulate claim sizes and their arrival times for Cramér-Lundberg model and first we analyze the probability that ruin happens in dependence on the parameters of our model for Pareto distributed claim size, next we compare ruin probability for other claim size distributions. For real life data we model the probability of large claim size occurence by generalized Pareto distribu- tion. 1
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