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Regression analysis of recurrent events
Rusá, Pavla ; Kulich, Michal (advisor) ; Komárek, Arnošt (referee)
V této práci se zabýváme metodami pro regresní analýzu výskytu opako- vaných událostí, při které je třeba se vypořádat se závislostí čas· do události v rámci jednoho subjektu. V první části práce se zabýváme možným rozšířením Coxova modelu proporcionálního rizika, který se využívá při analýze cenzoro- vaných dat, pro analýzu výskytu opakovaných událostí. Hlavní část práce je věnována odhadu parametr· v marginálních modelech a jejich asymptotickým vlastnostem. Následně se zabýváme i odhadem parametr· v marginálních mo- delech pro mnohorozměrná cenzorovaná data. Vhodnost použití marginálních model· je zkoumána pomocí simulací. 1
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Analysis of Variance with Random Effects
Hamerníková, Iva ; Komárek, Arnošt (advisor) ; Pešta, Michal (referee)
The aim of this thesis is to describe and derive the test of analysis of variance with random effects. At first we introduce a summary of results from the theory of probability which will be important in future derivations. Then we define the one-way classification model with fixed effects and propose the test statistics to test the equality of group means. In the following part we define the one-way classification model with random effects and derive properties of observations in this model. Under the assumption of balanced data we define sums of squares and derive their properties, which allow us to use them to create the test statistic. Finally we will use simulations in R to verify whether the ANOVA test with random effects observes the significance level when normality assumptions are violated.
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Multiple comparison with controls
Sychova, Maryna ; Hlávka, Zdeněk (advisor) ; Komárek, Arnošt (referee)
The main theme of the diploma thesis is description of multiple comparison methods, which are used to compare pairs of means or medians. At the beggining we define multiple testing and describe methods that control the probability of first type error at level α. The Šidák method and the prerequi- sites required for its use are described in detail. The work also includes a brief description of analysis of variance and an overview of several methods of multiple comparison. Additionally, the method of multiple comparison with control, its modifications and practical implementation is presented.
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Regularization and variable selection in regression models
Lahodová, Kateřina ; Komárek, Arnošt (advisor) ; Maciak, Matúš (referee)
This diploma thesis focuses on regularization and variable selection in regres- sion models. Basics of penalised likelihood, generalized linear models and their evaluation and comparison based on prediction quality and variable selection are described. Methods called LASSO and LARS for variable selection in normal linear regression are briefly introduced. The main topic of this thesis is method called Boosting. General Boosting algorithm is introduced including functional gradient descent, followed by selection of base procedure, especially the componentwise linear least squares method. Two specific application of general Boosting algorithm are introduced with derivation of some important characteristics. These methods are AdaBoost for data with conditional binomial distribution and L2Boosting for condi- tional normal distribution. As a final point a simulation study comparing LASSO, LARS and L2Boosting methods was conducted. It is shown that methods LASSO and LARS are more suitable for variable selection whereas L2Boosting is more fitting for new data prediction.
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Varying coefficient models
Sekera, Michal ; Maciak, Matúš (advisor) ; Komárek, Arnošt (referee)
The aim of this thesis is to provide an overview of the varying coefficient mod- els - a class of regression models that allow the coefficients to vary as functions of random variables. This concept is described for independent samples, longi- tudinal data, and time series. Estimation methods include polynomial spline, smoothing spline, and local polynomial methods for models of a linear form and local maximum likelihood method for models of a generalized linear form. The statistical properties focus on the consistency and asymptotical distribution of the estimators. The numerical study compares the finite sample performance of the estimators of coefficient functions. 1
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Tests for Paired Categorical Data
Míchal, Petr ; Komárek, Arnošt (advisor) ; Omelka, Marek (referee)
In this paper we deal with paired categorical data. We will test marginal ho- mogeneity and symmetry of corresponding probability table. At first, we describe multinomial distribution and contingency tables. In the next section, we deal with dichotomic paired categorical data, we derive McNemar's test and describe test for small sample sizes. Further, we state tests for general paired categorical data, Stuart's and Bhapkar's test are described. We then state test derived by Bowker, which is used for testing symmetry of probability table. In the last section, we show simulations of McNemar's test in software R. 1
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Bayesian variable selection
Jančařík, Joel ; Komárek, Arnošt (advisor) ; Hlávka, Zdeněk (referee)
The selection of variables problem is ussual problem of statistical analysis. Solving this problem via Bayesian statistic become popular in 1990s. We re- view classical methods for bayesian variable selection methods and set a common framework for them. Indicator model selection methods and adaptive shrinkage methods for normal linear model are covered. Main benefit of this work is incorporating Bayesian theory and Markov Chain Monte Carlo theory (MCMC). All derivations needed for MCMC algorithms is provided. Afterward the methods are apllied on simulated and real data. 1
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Homoscedasticity Tests in a Linear Model
Vávra, Jan ; Komárek, Arnošt (advisor) ; Hlávka, Zdeněk (referee)
This thesis deals with testing the assumption of homoscedasticity in linear model, that is the assumption of constant variance of this model. There is plenty of such tests, but not all of them can be applied to specific model and not all of them reach satisfactory results under various circumstances. Thesis focuses on tests which can be derived on the basis of the asymptotic theory for maximum likelihood estimation, particularly the test theory with nuisance parameters. There are derived two basic tests, the first one in the situation of analysis of variance model and the second one in the situation when we allow the dependence of variance to concomitant quantities. In subsequent numerical studies there are examined characteristics of derived test statistics. Powered by TCPDF (www.tcpdf.org)
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Analysis of variance when the assumption of homoscedasticity is violated
Zavadilová, Anna ; Omelka, Marek (advisor) ; Komárek, Arnošt (referee)
The method known as analysis of variance of simple sort offers a possibility of how to test equality of mean values of several random selections. At the same time, however, it requires the random selections to originate from normal distribution and to meet the condition of homoscedasticity, i.e. the requirement of identity of variances. The aim of this Thesis is to analyse consequences of violation of the assumptions of normality and homoscedasticity of the input data. The first part of the Thesis presents an overview of the course of the method based on the analysis of variance with standard assumptions. It is followed by the deriving of asymptotic distribution of test statistics, supposing the validity of the null hypothesis of identity of mean values in the case that neither the normality of input data nor the identity of variances is supposed. The findings are then applied to several special cases. The final part of the Thesis deals with a simulation study describing the influence of non-fulfilment of assumptions imposed on the test significance level. Powered by TCPDF (www.tcpdf.org)
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MCMC methods for financial time series
Tritová, Hana ; Pawlas, Zbyněk (advisor) ; Komárek, Arnošt (referee)
This thesis focuses on estimating parameters of appropriate model for daily returns using the Markov Chain Monte Carlo method (MCMC) and Bayesian statistics. We describe MCMC methods, such as Gibbs sampling and Metropolis- Hastings algorithm and their basic properties. After that, we introduce different financial models. Particularly we focus on the lognormal autoregressive model. Later we theoretically apply Gibbs sampling to lognormal autoregressive model using principles of Bayesian statistics. Afterwards, we analyze procedu- res, that we used in simulations of posterior distribution using Gibbs sampling. Finally, we present processed output of both simulated and real data analysis.
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