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Elementary statistical methods in psychological research
Benešová, Anna ; Omelka, Marek (advisor) ; Zichová, Jitka (referee)
This work deals with testing methods for a two-sample problem and their subsequent application to data from the field of behavioural economics. The main question of the analysis is to determine the existence of loss aversion and the en- dowment effect. This thesis presents tests suitable for analyzing the given data. Specifically, it focuses on testing the ratio and difference of means and the Mann-Whitney formulation for the Wilcoxon test. In the next sec- tion, the underlying article and the research topic are introduced. In the end, the methods are applied to the provided data set. The existence of the endowment effect is confirmed in at least half of the cases.
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Theoretical and empirical quantiles and their use for prediction interval construction
Šimičák, Jakub ; Maciak, Matúš (advisor) ; Omelka, Marek (referee)
The purpose of the bachelor thesis is to introduce the reader to two approaches to the construction of prediction intervals. The first procedure assumes a probabilistic model and leads to a frequentist prediction interval that uses the relevant theoretical quantiles of probability distributions. The second procedure assumes no probabilistic model and leads to a conformal prediction interval that uses empirical quantiles of the relevant random sample. In the course of the paper, both approaches will be derived in general terms and then illustrated with concrete examples. The thesis also includes a simulation study comparing the empirical coverage of frequentist and conformal prediction inter- vals for random selections from different distributions. 1
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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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