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Behrens-Fisher Problem
Kurková, Michaela ; Jurečková, Jana (advisor) ; Kalina, Jan (referee)
In this work we are concerned with the problem of testing the equality of the means from the two populations for the case where the population covariance matrices are unequal. Well-known problem frequently occurring in applied statistics, called the Behrens-Fisher problem. This is widely used in economy, social sciences and medicine. We concentrate mainly on multivariate case. We study variety of possible approaches, bayesian, parametric, nonparametric, with particular solutions. At the close we mention one of possible solutions of generalized multivariate Behrens-Fisher problem, Mack-Wolfe test. A Monte Carlo simulation was conducted in order to illustrate their properties for different dimensions, the degree of heteroscedasticity, sample sizes and correlation coefficients. For comparison we mention classical two sample multivariate Hotelling T2. We demonstrate impact of nonnormality by estimating the type I error and power for selected solutions. Finally we use real data form Forest inventory in the Czech Republic 2001-2004 and we present the necessity of solution of this problem in practice.
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Two-step statistical procedures
Rusá, Šárka ; Hušková, Marie (advisor) ; Jurečková, Jana (referee)
Title: Two-step statistical procedures Author: Šárka Rusá Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Marie Hušková, DrSc., Department of Probability and Mathematical Statistics Abstract: The Bachelor thesis deals with specific sequential confidence inter- val estimation and hypothesis testing. Initially, we limit ourselves to the study of a random sample selected from a Normal population. Chapter 1 is devoted to the construction of a fixed-width confidence interval for the mean given by Stein's two-step procedure for an unknown variance. In Chapter 2 with the use of a theorem proven in Chapter 1 it is possible to test a hypothesis for a mean value while the type II error remains independent of the variance. In the third chapter we present a modification of Stein's procedure, whose motivation consists in the reduction in the mean of the final sample size. The generalization of the modified procedure, which is applicable to distributions with an unknown finite non-zero variance, is treated in Chapter 4. In Chapters 1 and 4 we will simulate the distribution of the random variable which determines the final sample size. Keywords: Sequential estimation, Stein's two-step procedure, fixed-width confidence intervals.
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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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Paired comparisons in ANOVA
Hrušková, Iveta ; Omelka, Marek (advisor) ; Jurečková, Jana (referee)
The problem of testing multiple hypotheses at once is called the problem of multiple testing. We focused on comparing more than two means in one- way analysis of variance, also known as ANOVA. We dealt with the Tukey me- thod, the Hothorn-Bretz-Westfall method, the bootstrap-based methods and also the Bonferroni method and its modification by the Holm method, the last two methods being popular mainly for their simplicity. We focused in detail on the asymptotic behavior of these methods and then compared them using si- mulations in terms of compliance with the prescribed level and in terms of average strength. Bonferroni's method, which is conservative, is known to lose strength compared to other methods. However, its modification of Holm's method, which is also conservative, in some cases by its strength equates to other more complex methods. 1
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Robustness of statistical estimates
Hobza, Vojtěch ; Maciak, Matúš (advisor) ; Jurečková, Jana (referee)
The bachelor thesis deals with the aspect of robustness of estimates, everything is dis- cussed in detail on M-estimates. However, the first chapter explains in detail the concept of robustness and its characteristics, which can be used to measure the robustness. The second chapter then deals directly with the theory of M-estimates and their use in estimating the location parameter. Both robust examples of M-estimates (sample median, Huber's estimate, Tukey's estimate) and non-robust representatives of this group (sample mean) are given here. In the third chapter, the difference between robust and non-robust estimates is illustra- ted on the simulated data. It is shown that in some cases a non-robust estimate can be used, but in other situations such a non-robust estimate can fail completely. 1
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Special issue of the Conference Analytical Methods in Statistics (AMISTAT 2019)
Kalina, Jan ; Jurečková, Jana
IN: Applications of Mathematics. 2020, 65(3), 227-342. ISSN 0862-7940. doi: 10.21136/AM.2020.0106-20. ANNOTATION: This special issue of Applications of Mathematics is devoted to the third workshop on Analytical Methods in Statistics (AMISTAT 2019), which took place in Liberec on September 16–19, 2019. It was organized by the Department of Applied Mathematics at the Faculty of Science, Humanities and Education of the Technical University of Liberec. The workshop was held under the auspices of Miroslav Brzezina, Rector of the Technical University of Liberec.\n
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Order statistics
Tělupil, Dominik ; Hlávka, Zdeněk (advisor) ; Jurečková, Jana (referee)
The aim of this work is to introduce general notions of theory of order statistics and the method of ranked set sampling (RSS). This statistical method is based on ranking of observed random variables which allows us to con- struct an unbiased estimator with lower variance than by using the method of sim- ple random sampling. In this work we will study estimators of the expected va- lue. Furthermore, we will investigate some properties of estimators based on RSS and some modifications of this method. The thesis contains a chapter about soft- ware simulations which verify the properties of estimators based on RSS. 1
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Cross-entropy based combination of discrete probability distributions for distributed decision making
Sečkárová, Vladimíra ; Kárný, Miroslav (advisor) ; Jurečková, Jana (referee) ; Janžura, Martin (referee)
Dissertation abstract Title: Cross-entropy based combination of discrete probability distributions for distributed de- cision making Author: Vladimíra Sečkárová Author's email: seckarov@karlin.mff.cuni.cz Department: Department of Probability and Mathematical Statistics Faculty of Mathematics and Physics, Charles University in Prague Supervisor: Ing. Miroslav Kárný, DrSc., The Institute of Information Theory and Automation of the Czech Academy of Sciences Supervisor's email: school@utia.cas.cz Abstract: In this work we propose a systematic way to combine discrete probability distributions based on decision making theory and theory of information, namely the cross-entropy (also known as the Kullback-Leibler (KL) divergence). The optimal combination is a probability mass function minimizing the conditional expected KL-divergence. The ex- pectation is taken with respect to a probability density function also minimizing the KL divergence under problem-reflecting constraints. Although the combination is derived for the case when sources provided probabilistic type of information on the common support, it can applied to other types of given information by proposed transformation and/or extension. The discussion regarding proposed combining and sequential processing of available data, duplicate data, influence...
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Wilcoxon two-sample test
Šlampiak, Tomáš ; Omelka, Marek (advisor) ; Jurečková, Jana (referee)
In this work we present two-sample Wilcoxon statistic test. It is explained which parameter of data could be tested. Using knowledge of projection of random variables there is shown the derivation of the asymptotic distribution under the null hypothesis. We derive two versions of test, first of them supposes a shift model, while the other one just assumes that observations consist of two independent samples with continuous distributions. Finally, we deduce the behaviour of both tests from simulations, especially the impact on the first version when there is assumed no shift model.
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