
Zero inflated Poisson model
Veselý, Martin ; Komárek, Arnošt (advisor) ; Hlávka, Zdeněk (referee)
This paper deals with the zeroinflated Poisson distribution. First the Poisson model is defined and generalized to a zeroinflated model. The basic properties of this generalized model are derived. After wards the basics of the method of moments and the maximum likelihood method are described. Both of these are used to derive parameter estimates of such distribution. The feasibility of calculating the distribution of moment method estimates is analyzed. Then the asymptotic distribution of maximum likelihood estimates is derived and used to create confidence intervals. In the last chapter a numeric si mulation of the derived asymptotic properties is performed. Special attention is paid to situations where regularity conditions are not met. 1


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 socalled variancestabilizing 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.


Sample Quantiles
Hrušková, Iveta ; Komárek, Arnošt (advisor) ; Nagy, Stanislav (referee)
If the distribution of random variable is uknown, we are not able to figure out the value of theoretical quantile. In case there is a random sample from this distribution, it is possible to estimate the value of theoretical quantile. This es timation is called sample quantile. This work is focused on nine frequently used varieties of sample quantile. They will be compared by means of characteristics that can be examined when speaking about sample quantile. All these varieties will be demonstrated on simple example. Finally, there will be shown that all these versions of sample quantile are consistent estimators of theoretical quan tile. The construction of confidence interval for theoretical quantile will be the topic of the final part of the work. 1


Statistical tests in stratified fourfold tables
Vook, Peter ; Komárek, Arnošt (advisor) ; Omelka, Marek (referee)
This paper deals with statistical tests in stratified fourfold tables. Several tests of conditional indepen dence are derived in it. A test of homogeneous association is also described. At first, contingency tables with arbitrary dimensions and multinomial distribution are defined. Then we continue with a description of fourfold tables and their binomial representation. In the next section we deal with an odds ratio and its asymptotic distribution. Formal definition of stratification and relevant terms follows afterwards. In the next chapter a derivation of test statistics for conditional independence tests including the wellknown CochranMantelHaenszel test based on a hypergeometric distribution can be found. This chapter also includes a description of BreslowDay test of homogeneous association. A numerical simulation of chosen tests is performed eventually. 1


Prediction error for mixed models
Šlampiak, Tomáš ; Komárek, Arnošt (advisor) ; Hlávka, Zdeněk (referee)
A Linear mixedeffects model (LME) is one of the possible tools for longitudinal or groupdependent data. This thesis deals with evaluating of prediction error in LME. Firstly, it is derived the mean square error of prediction (MSEP) by direct calculation. Then the covariance penalty method and crossvalidation is presented for evaluation of MSEP in LME. Further, it is shown how Akaike information criterion (AIC) can be used in mixedeffects models. Because of the model's properties two types of AIC are distinguished  marginal and conditional one. Subsequently, the procedures of AIC's calculation and its basic asymptotic properties are described. Finally, the thesis contains simulation study of behaviour of marginal and conditional AIC with the goal to choose the right variance structure of random effects. It turns out that the marginal criterion tends to select models with smaller number of random effects than conditional criterion.


Bayesian factor analysis
Vávra, Jan ; Komárek, Arnošt (advisor) ; Maciak, Matúš (referee)
Bayesian factor analysis  abstract Factor analysis is a method which enables highdimensional random vector of measurements to be approximated by linear combinations of much lower number of hidden factors. Classical estimation procedure of this model lies on the cho ice of the number of factors, the decomposition of variance matrix while keeping identification conditions satisfied and on the appropriate choice of rotation for better interpretation of the model. This model will be transferred into bayesian framework which offers the usage of prior information unlike the classical appro ach. The number of hidden factors can be considered as a random parameter and the dependency of each measurement on at most one factor can be forced by suitable specification of prior distribution. Estimates of model parameters are based on posterior distribution which is approximated by Monte Carlo Markov Chain methods. Bayesian approach solves the problem of selection of the num ber of factors, the model estimation and the ensuring of the identifiability and the interpretability at the same time. The ability to estimate the real number of hidden factors is tested in a simulation study. 1


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


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 oneway classification model with fixed effects and propose the test statistics to test the equality of group means. In the following part we define the oneway 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.


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.


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.
