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Statistical tests of normality
Krupa, Tomáš ; Maciak, Matúš (advisor) ; Omelka, Marek (referee)
The aim of this paper is to present the well-known normality tests used in practice and to compare them. The first chapter consists of the basic concepts and properties of the nor- mal distribution. In the second chapter 6 normality tests are treated, namely Kolmogorov- Smirnov, Lilliefors, Shapiro-Wilk, Anderson-Darling, D'Agostino-Pearson and Jarque- Bera. For each test, test statistic and shape of critical region are given, among others. The third chapter, with empirical study, contains two parts. In the first part, nature of the study is briefly explained and level of significance declared by tests is empirically-checked. In the second part, power of tests is empirically compared against various alternatives and the results are discussed. 1
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Delta method and its generalizations
Pavlech, Ján ; Omelka, Marek (advisor) ; Nagy, Stanislav (referee)
The goals of this thesis are various generalizations of the classical delta theorem, in which the advantage is that we can separately investigate the analytical properties of transformation of the estimate, and independently, we can deal with asymptotic properties of the original estimate. When working with Euclidean spaces, we generalize the delta theorem for the case that partial derivatives are not continuous or they are equal to zero. When working with general normed linear spaces, we first examine Hadamard- differentiability, while formulating and proving equivalence with Fréchet-differentiability, under proper assumptions. We demonstrate the functional delta theorem on known results for empirical quantiles and median absolute deviation in the case of a random sample, together with our own result for the interquartile range and empirical quantiles in the case of AR(d) sequence. We also show why the functional delta theorem is not usable for moment estimators. In the last part, we examine the Hadamard-differentiability of a copula functional and its application to the derivation of the asymptotic distribution of the empirical copula. 1
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Large dimension of regressors in regression problems
Semjonov, Valerij ; Omelka, Marek (advisor) ; Mizera, Ivan (referee)
This thesis deals with asymptotic properties of least squares estimators of regression coefficients of linear models with a large dimension of regressors. Particularly, consistency and asymptotic normality are investigated. Several types of consistency are defined and their mutual relation is discussed. Theorems on the asymptotic normality are formulated separately for random and fixed designs. The average proportion of the cases when the true regression coefficients are not covered by the asymptotic confidence intervals is calculated for some chosen linear models by means of simulations. Specifically, for One- way ANOVA these average proportions are compared with the theoretical probabilities given by the derived asymptotic formulae. 1
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Tolerance limits
Bedřich, Marek ; Omelka, Marek (advisor) ; Komárek, Arnošt (referee)
This bachelor's thesis deals with tolerance intervals, a statistical tool used to quan- tify the uncertainty of statistical predictions. The introductory part of the text briefly recalls confidence intervals. The thesis then focuses on prediction intervals, which are an intermediate step between confidence intervals and tolerance intervals. Specifically, the prediction interval for normal distribution and nonparametric prediction interval are analyzed. The main part of the thesis then deals with tolerance intervals - the definition, construction of both parametric and nonparametric tolerance intervals, convergence, and actual coverage of the derived intervals. In the final part, an example of the practical application of this tool is presented. 1
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Two-dimensional distributions for given margins
Šťastný, Filip ; Pešta, Michal (advisor) ; Omelka, Marek (referee)
One of the tools for study of dependence between random variables are co- pulas. While modelling multidimensional variables it is possible using Sklar's theorem to model through copulas marginal distributions and relationship be- tween them separately, this approach thus enables us to split construction of multi-dimensional distributions into these two factors. With marginal distributi- ons fixed, the construction is consisting of appropriate copula choice only. This thesis deals with copulas in the case of two-dimensional distributions with conti- nuous fixed marginal distributions and is focused on parametrical copulas, mainly through Archimedean copulas. Basic properties of copulas with Sklar's theorem, which enables studying copulas in stochastic context, are presented here. Further, measures of dependence such as Kendall's tau, Spearman's rho and coeficients of tail dependence are in connection with copulas studied in this thesis. At the end, the thesis deals with methods of estimation unknown parameters, which are ilustrated on two examples. 1
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Copula based models for multivariate time series
Šír, David ; Hudecová, Šárka (advisor) ; Omelka, Marek (referee)
The thesis deals with the modelling of multivariate time series. The SCOMDY model is described. It models individual univariate time series using an ARMA-GARCH, and their dependence structure is modelled using a copula. For copula selection goodness-of- fit test is discussed. Predictions are presented with algorithms for constructing prediction intervals. The whole theory is demonstrated with examples. Monte Carlo simulations verify the suitability and applicability of the theory. The SCOMDY model is applied to a three-dimensional time series consisting of the closing prices of stocks of Apple Inc. Microsoft Corporation and Alphabet Inc. 1
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Chebyschev type inequalities
Vachálek, Vladimír ; Hlubinka, Daniel (advisor) ; Omelka, Marek (referee)
Title: Chebyschev type inequalities Author: Vladim'ır Vach'alek Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Daniel Hlubinka, Ph.D., Department of Probability and Mathematical Statistics Abstract: In the presented thesis we deal with Chebyshev type inequalities for bounded random variables. In the first chapter we introduce and prove Hoeffding, Bennett and Bernstein inequalities and explain some relationships. In the second chapter we show how tight are the estimates given by each inequality compared to true probability and to the estimate given by central limit theorem on four distributions with graphical processing of results. Keywords: Chebyshev inequality, Hoeffding inequality, Bennett inequality, Bern- stein inequality 1
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Regression depth and related methods
Dočekalová, Denisa ; Nagy, Stanislav (advisor) ; Omelka, Marek (referee)
While the halfspace depth has gained more and more popularity in the recent years as a robust estimator of the mean, regression depth, despite being based on a similar concept, is still a relatively unknown method. The main goal of this paper was therefore to introduce the concept of robust depth to the reader, illustrate its geometric interpre- tation, and provide at least a basic overview of the findings that occurred within the individual researches. Finally, a small simulation study was conducted comparing the de- epest regression method with other selected methods commonly used in practice, namely the method of least absolute deviations and ordinary least squares method. 1
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Tests of independence in contingency tables
Gažová, Miroslava ; Omelka, Marek (advisor) ; Vávra, Jan (referee)
This thesis deals with the problem of independence testing between two discrete ran- dom variables. At first, we define contingency table and the basic notations in the context of independence tests. We describe the most commonly used tests in this field. Next, we present the U-statistics Permutation test of independence (USP), which was first presen- ted by authors T.B.Berrett and R.J.Samworth (2021). In the next section, we focus in better detail on fourfold contingency tables and corresponding problem of testing for the equality of parameters from two independent binomial distributions. In the end, we apply the tests on the real data using the R enviroment. 1
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Uniform law of large numbers, VC dimension and machine learning
Kossumov, Aibat ; Omelka, Marek (advisor) ; Týbl, Ondřej (referee)
In this thesis we study the generalized Glivenko-Cantelli theorem and its application in mathematical foundations of machine learning. Firstly, we prove the generalized Glivenko-Cantelli's theorem using covering numbers and lemma of symmetrization. Next we show the uniform law of large numbers. Then, we deal with Vapnik-Chervonenkis classes of functions (VC classes). We show that for VC classes covering numbers are uniformly bounded. Finally, we describe the task of machine learning and give an example of one specific task that can be "learned". The main application will be to prove the fundamental theorem of statistical learning. Usually this theorem is proved for classes of predictors that are Probably Approximately Correct learnable (PAC learnable). In this work we strengthen the property of PAC learnable and for it we prove the basic theorem of statistical learning. 1
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