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Generalized estimating equaitons
Sotáková, Martina ; Omelka, Marek (advisor) ; Antoch, Jaromír (referee)
In this thesis we are interested in generalized estimating equations (GEE). First, we introduce the term of generalized linear model, on which generalized estimating equations are based. Next we present the methos of pseudo maximum likelyhood and quasi-pseudo maximum likelyhood, from which we move on to the methods of generalized estimating equations. Finally, we perform simulation studies, which demonstrates the theoretical results presented in the thesis. 1
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Two-sample Wilcoxon test in the presence of ties
Kulla, Filip ; Omelka, Marek (advisor) ; Maciak, Matúš (referee)
This work is devoted to the Wilcoxon rank-sum test in the presence of ties. In this work, asymptotic distribution of the Wilcoxon test statistic in the presence of ties is derived using well-known results about U-statistics. With the aid of this result, a corrected test statistic for data containing ties is proposed. Furthermore, the thesis examines the relation between the derived correction and conditional variance of the test statistic (and conditional expectation). Finally, by means of simulation, the effect of the derived correction on the actual significance level is examined as the number of tied observations increases. 1
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Joint Models for Longitudinal and Time-to-Event Data
Vorlíčková, Jana ; Komárek, Arnošt (advisor) ; Omelka, Marek (referee)
Title: Joint Models for Longitudinal and Time-to-Event Data Author: Jana Vorlíčková Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Arnošt Komárek, Ph.D., Department of Probability and Mathematical Statistics Abstract: The joint model of longitudinal data and time-to-event data creates a framework to analyze longitudinal and survival outcomes simultaneously. A commonly used approach is an interconnection of the linear mixed effects model and the Cox model through a latent variable. Two special examples of this model are presented, namely, a joint model with shared random effects and a joint latent class model. In the thesis we focus on the joint latent class model. This model assumes an existence of latent classes in the population that we are not able to observe. Consequently, it is assumed that the longitudinal part and the survival part of the model are independent within one class. The main intention of this work is to transfer the model to the Bayesian framework and to discuss an estimation procedure of parameters using a Bayesian statistic. It consists of a definition of the model in the Bayesian framework, a discussion of prior distributions and the derivation of the full conditional distributions for all parameters of the model. The model's ability to...
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Testing independence in two-by-two tables
Obukhov, Andrey ; Omelka, Marek (advisor) ; Kulich, Michal (referee)
The main purpose of this work is to describe three well-known statistical tests of independence in two-by-two contingency tables. We will deeply study chi- squared test of independence, Fisher's exact test and Barnard's test and apply them on examples. Also we will describe, in general, categorical variables, which are often analysed using a multinomial distribution. At the end we will apply tests on the examples, using data simulated from a multinomial and binomial distribution. 1
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Testing independence in two-by-two tables
Obukhov, Andrey ; Omelka, Marek (advisor) ; Kulich, Michal (referee)
The main purpose of this work is to describe three well-known statistical tests of independence in two-by-two contingency tables. We will deeply study chi- squared test of independence, Fisher's exact test and Barnard's test and apply them on examples. Also we will describe, in general, categorical variables, which are often analysed using a multinomial distribution. At the end we will apply tests on the examples, using data simulated from a multinomial and binomial distribution. 1
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Aggregation of dependent risks
Asipenka, Anna ; Mazurová, Lucie (advisor) ; Omelka, Marek (referee)
In this thesis we are interested in the calculation of economic capital for the to- tal loss which is the sum of partial dependent losses, whose dependence structure is described by Archimedean and hierarchical Archimedean copulas. Firstly, the concept of economic capital and the ways of its aggregation are introduced. Then the basic definitions and properties of copulas are listed, as well as the depen- dence measures. After that we work with definition and properties of Archimedean copulas and their simulation. We also mention the most popular families of Ar- chimedes copulas. Next, hierarchical Archimedean copulas are defined, as well as the algorithm for their sampling. Finally, we present methods for estimating the parameters of copulas and the recursive algorithm for estimating the hierarchical Archimedean copula structure. In the last chapter we perform simulation studies of selected models using hierarchical Archimedes copulas. 1
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Edgeworth expansion
Dzurilla, Matúš ; Omelka, Marek (advisor) ; Nagy, Stanislav (referee)
This thesis is focused around Edgeworth's expansion for approximation of distribution for parameter estimation. Aim of the thesis is to introduce term Edgeworth's expansion, its assumptions and terminology associated with it. Afterwards demonstrate process of deducting first term of Edgeworth's expansion. In the end demonstrate this deduction on examples and compare it with different approximations (mainly central limit theorem), and show strong and weak points of Edgeworth's expansion.
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Variance stabilizing transformations
Kuželová, Noemi ; Omelka, Marek (advisor) ; Komárek, Arnošt (referee)
Abstract. We often examine data whose sample mean converges to a normal distribution, but the variance generally depends on an unknown parameter. To get rid of this dependence, we can sometimes use the so-called variance-stabilizing transformation method. Firstly, this thesis explains the method in detail and finds a general procedure to find suitable transformations. Then it will focus on data from Poisson and binomial distributions with unknown parameters. For these data, it finds transformations that stabilize (asymptotic) variance, and compares them with the "improved"transforms from the article Anscombe (1948). Most of the thesis is devoted to the shape of these transformations. Finally, we show in the Poisson distribution simulation that it is really appropriate to use this method and compare the derived transformation with its Anscombe version.
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Continuity correction
Štěpán, Marek ; Omelka, Marek (advisor) ; Maciak, Matúš (referee)
For an approximation of discrete random variable, which is the sum of n inde- pendent, identically distributed discrete random variables, we can use the central limit theorem. However, it turns out that we can refine this approximation by applying continuity correction. This term is explained in the thesis, and it is illustrated several ways how the continuity correction can be derived. There is also a numerical comparison of the approximation error for the binomial distribu- tion approximation by the normal distribution with the correction for continuity and approximation without the correction. There are also described confidence intervals and χ2 test of independence in contingency tables in which continu- ity correction are used. On simulations for various parameters, we will test the properties of these intervals (true confidence level and length) and tests (actual significance level and power).
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Score tests in contingency tables
Jex, Martin ; Omelka, Marek (advisor) ; Kulich, Michal (referee)
The thesis deals with testing of hypotheses in multinomial distribution. It utilizes two approaches, Pearson's approach known as the of goodness of fit test and the approach stemming from theory of maximum likelihood. The thesis presents derivations of tests based on maximum likelihood. Both approaches are used on the multinomial distribution and for both cases with and without nuisance parameters. The links between both approaches are presented as well. Furthermore both approaches are illustrated on real data to facilitate better understanding of the discussed problems. 1
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