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Examination of k regression lines
Drozen, Alan ; Zvára, Karel (advisor) ; Omelka, Marek (referee)
In the present work we study the problem of k regression lines in the general linear model. First we describe the general linear model with a multivariate normal distribution of errors and we show some of its basic characteristics. Then we introduce a model with k regression lines. Further, we describe a test for testing the hypothesis of two regression lines being parallel and another one for testing all or some of the k regression lines being parallel or identical. Then we derive the test of the submodel of the general linear model and analyze issues such as the power of this test, the submodel of another submodel, the orthogonality and reparametrization. We show geometric interpretations of the general linear model and of the submodel test as well. In the subsequent part, we focus on nonparametric tests. We present four permutation tests for testing the submodel in the general linear model. Finally we perform numerical simulation to find out whether the tests match the required size and to determine their power.
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Confidence intervals for parameters of multinomial distribution
Bárnetová, Kamila ; Anděl, Jiří (advisor) ; Omelka, Marek (referee)
Title: Confidence intervals for parameters of multinomial distribution Author: Kamila Bárnetová Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Jiří Anděl, DrSc., Department of Probability and Mathematical Statistics Abstract: Confidence intervals for parameters for binomial and multinomial distribution are described in this thesis. These intervals can be used in practice, for exemple- pre-election estimates. The first two chapter are devoted to derivation of these intervals. Simulations and comparison of several selected methods can be found in the last chapter. Based on the simulations, we consider it appropriate, to choose to calculate confidence intervals for parameters of multinomial distribution intervals based on Bonferroniho inequality, or their modifications. These intervals are easy to calculate, while their coverage probability is at least 0.89. Keywords: confidence interval, multinomial distribution, binomial distribution, Bonferroni inequality
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Poisson Approximations
Klikáč, Jan ; Omelka, Marek (advisor) ; Kulich, Michal (referee)
This bachelor thesis deals with the counting probability using Poisson distri- bution and shows new ways of approximating Poisson distribution. The first chapter summarizes the findings regarding the Poisson distribution, its definition and properties. It also show a limit transition from the binomial distribution to Poisson distibution and examples demonstrating the usage of this limit transition. Brun Sieve is introduced in the second chapter. It gives a new possibility of transiting to a Poisson distribution. Random variables, which we want to appro- ximate, no longer need to have binomial distribution. Instead the property of expected value is required. The second part of the chapter includes a practical demonstration of the usage of Brun Sieve. In the third chapter we estimate size of the error that we made when approxi- mating to Poisson distribution. There is also formulated Stein-Chen theorem for estimating the error of Poisson approximation and its version for a special case. Keywords: Poisson distribution, Brun Sieve, Stein-Chen theorem 1
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Parameter Estimation under Two-phase Stratified and Cluster Sampling
Šedová, Michaela ; Kulich, Michal (advisor) ; Picek, Jan (referee) ; Omelka, Marek (referee)
Title: Parameter Estimation under Two-phase Stratified and Cluster Sampling Author: Mgr. Michaela Šedová Department: Department of Probability and Mathematical Statistics Supervisor: Doc. Mgr. Michal Kulich, Ph.D. Abstract: In this thesis we present methods of parameter estimation under two-phase stratified and cluster sampling. In contrast to classical sampling theory, we do not deal with finite population parameters, but focus on model parameter inference, where the ob- servations in a population are considered to be realisations of a random variable. However, we consider the sampling schemes used, and thus we incorporate much of survey sampling theory. Therefore, the presented methods of the parameter estimation can be understood as a combination of the two approaches. For both sampling schemes, we deal with the concept where the population is considered to be the first-phase sample, from which a sub- sample is drawn in the second phase. The target variable is then observed only for the subsampled subjects. We present the mean value estimation, including the statistical prop- erties of the estimator, and show how this estimation can be improved if some auxiliary information, correlated with the target variable, is observed for the whole population. We extend the method to the regression problem....
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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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