
Kernel Methods in Particle Filtering
Coufal, David ; Beneš, Viktor (advisor) ; Klebanov, Lev (referee) ; Studený, Milan (referee)
Kernel Methods in Particle Filtering David Coufal Doctoral thesis  abstract The thesis deals with the use of kernel density estimates in particle filtering. In particular, it examines the convergence of the kernel density estimates to the filtering densities. The estimates are constructed on the basis of an out put from particle filtering. It is proved theoretically that using the standard kernel density estimation methodology is effective in the context of particle filtering, although particle filtering does not produce random samples from the filtering densities. The main theoretical results are: 1) specification of the upper bounds on the MISE error of the estimates of the filtering densities and their partial derivatives; 2) specification of the related lower bounds and 3) providing a suitable tool for checking persistence of the Sobolev character of the filtering densities over time. In addition, the thesis also focuses on designing kernels suitable for practical use. 1


Stable distributions and their applications
Volchenkova, Irina ; Klebanov, Lev (advisor) ; Beneš, Viktor (referee)
The aim of this thesis is to show that the use of heavytailed distributions in finance is theoretically unfounded and may cause significant misunderstandings and fallacies in model interpretation. The main reason seems to be a wrong understanding of the concept of the distributional tail. Also in models based on real data it seems more reasonable to concentrate on the central part of the distribution not tails. Powered by TCPDF (www.tcpdf.org)


Normality test of the gene expression data
Shokirov, Bobosharif ; Klebanov, Lev (advisor) ; Hušková, Marie (referee) ; Kalina, Jan (referee)
This thesis deals with a test of normality of gene expressions data. Based on characterization theorems of the normal distribution, the test of normality is replaced by a test of spherical uniformity. Due to strong correlations between the gene expression data, the normality test is conducted with $\delta$ sequences. A new characterization theorem of the normal distribution is proven. Based on that, the normality test is conducted using Kolmogorov's test statistic. The obtained characterization results for the normal distribution are extended to the complete type of distributions and based on that, a test is conducted to verify whether the distributions of the two data sets of the gene expressions belong to the same type. Powered by TCPDF (www.tcpdf.org)


Linear forms and characterization of probability distributions
Božoňová, Denisa ; Klebanov, Lev (advisor) ; Prášková, Zuzana (referee)
In this paper we will discuss the characterization of strictly νnormal and strictly νstable distribution. At the beginning we mentioned some basic concepts, which we then use in this work. Such as, strongly monotone operator, strictly ξpositive family, or linear form. Further, we describe the characterization of strictly νnormal and strictly νstable distribution using the above definitions. We also lists examples of νstable distributions and we prove corresponding results. In the last chapter we look at the use of mentioned distributions in practice, namely νstable distribution. 1


Extremal measures in probability
Kešelj, Sonja ; Dostál, Petr (advisor) ; Klebanov, Lev (referee)
Pólya urn scheme is a parametric probability model with interesting characteristics, which we shall look into within the scope of this thesis. Furthermore, using Bayesian approach we will show that, under certain conditions, the aforementioned model is equivalent to the Bernoulli scheme of independent alternative trials with random parameter that has beta distribution. Another subject of the thesis is ergodic theory of stationary sequances, as well as extremal analysis of probability measures that are invariant under some measurable transformation. This is illustrated on an example of homogegenous Markov chain with stationary distribution. The final segment of the thesis focuses on the theory of financial derivatives pricing  more specifically, finding arbitragefree price using martingale measures. To this we add examples of application on binomial pricing trees. Keywords: extremal measure, Pólya urn scheme, ergodic and stationary sequences, financial derivatives pricing Powered by TCPDF (www.tcpdf.org)


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, pvalues 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 twosample KolmogorovSmirnov test and interesting behav ior of Hotelling's test for dependent components of observations. In the end of...


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


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)


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


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 Hfunctions. 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...
