
Basic stochastic epidemic models
Strachoňová, Karla ; Hudecová, Šárka (advisor) ; Kulich, Michal (referee)
This thesis deals with two basic models which are used for epidemic model ling in closed populations, namely Greenwood and ReedFrost models. At first, knowledge which a reader needs to have about Markov chains and random varia bles is summarized. Then the two models are described by modelling the number of susceptible and infectious individuals, as well as the duration and size of the epidemic. All of these approaches to modelling an epidemic are then illustrated on examples. Finally, the maximum likelihood method of the probability of infection is described and illustrated on real data in the last chapter, where the obtained results are discussed as well. 1


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


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


Confidence intervals for differences and ratios of proportions
Krnáč, Ľuboš ; Kulich, Michal (advisor) ; Zvára, Karel (referee)
The Bachelor thesis deals with the creation of confidence intervals for diffe rence of parameters of two distributions. In the first part we consider the problem of making such confidence intervals for differences. Then we try to find sufficient conditions for MOVER, which leads to new, nontrivial confidence intervals for difference of parameters of two distributions. These confidence intervals have im proved and desired properties. There are also examples of usage of MOVER, and possible difficulties. The third section contains graphs of coverage probabili ties for different input intervals. These graphs are made to show different levels of achieved coverage probabilities for some input intervals, namely ClopperPearson, Wald, Wilson and logit. 1


Parameter estimation of gamma distribution
Zahrádková, Petra ; Kulich, Michal (advisor) ; Hlávka, Zdeněk (referee)
It is wellknown that maximum likelihood (ML) estimators of the two parame ters in a Gamma distribution do not have closed forms. The Gamma distribution is a special case of a generalized Gamma distribution. Two of the three likeli hood equations of the generalized Gamma distribution can be used as estimating equations for the Gamma distribution, based on which simple closedform estima tors for the two Gamma parameters are available. Intuitively, performance of the new estimators based on likelihood equations should be close to the ML estima tors. The study consolidates this conjecture by establishing the asymptotic beha viours of the new estimators. In addition, the closedforms enable biascorrections to these estimators. 1


Statistical Methods for Regression Models With Missing Data
Nekvinda, Matěj ; Kulich, Michal (advisor) ; Omelka, Marek (referee)
The aim of this thesis is to describe and further develop estimation strategies for data obtained by stratified sampling. Estimation of the mean and linear regression model are discussed. The possible inclusion of auxiliary variables in the estimation is exam ined. The auxiliary variables can be transformed rather than used in their original form. A transformation minimizing the asymptotic variance of the resulting estimator is pro vided. The estimator using an approach from this thesis is compared to the doubly robust estimator and shown to be asymptotically equivalent.


Postselection Inference: Lasso & Group Lasso
Bouř, Vojtěch ; Maciak, Matúš (advisor) ; Kulich, Michal (referee)
The lasso is a popular tool that can be used for variable selection and esti mation, however, classical statistical inference cannot be applied for its estimates. In this thesis the classical and the group lasso is described together with effici ent algorithms for the solution. The key part is dedicated to the postselection inference for the lasso estimates where we explain why the classical inference is not suitable. Three postselection tests for the lasso are described and one test is proposed also for the group lasso. The tests are compared in simulations where finite sample properties are examined. The tests are further applied on a practical example. 1


Confidence Intervals for Quantiles
Horejšová, Markéta ; Kulich, Michal (advisor) ; Hlávka, Zdeněk (referee)
In this thesis, various construction methods for simultaneous confidence intervals for quantiles are explained. Among nonparametric approaches, a special emphasis is dedicated to a recent method based on a multinomial distribution for calculating the overall confidence level of confidence intervals for all quantiles of interest using an efficient recursive algorithm, which is also described. Furthermore, a method based on KolmogorovSmirnov statistic or an asymptotic method using empirical distribution function and order statistics for quantile estimate are presented. A special parametric method for several quantiles of a normally distributed population is introduced along with a few of its modifications. Subsequently, a simulation is run to test the real coverage of the described theoretical methods. Powered by TCPDF (www.tcpdf.org)


Statistical analysis of datasets with missing observations
Janoušková, Kateřina ; Omelka, Marek (advisor) ; Kulich, Michal (referee)
Mechanisms of missing data and methods are described in this thesis. Three mechanisms are considered  MCAR, MAR, MNAR. Two simple methods using deletion of incomplete records are shown and their properties and shortcomings are demonstrated. Secondly, the principle of simple imputations is explained. EM algorithm which uses the classical statistics and the algorithm of data augmentation which uses Bayesian framework are derived and compared. The last method included in the thesis is the multiple imputation. The described methods are compared on real data set, first on continuous variables and then on a contingency table. 1


Models for zeroinflated data
Matula, Dominik ; Kulich, Michal (advisor) ; Hlubinka, Daniel (referee)
The aim of this thesis is to provide a comprehensive overview of the main approaches to modeling data loaded with redundant zeros. There are three main subclasses of zero modified models (ZMM) described here  zero inflated models (the main focus lies on models of this subclass), zero truncated models and hurdle models. Models of each subclass are defined and then a construction of maximum likelihood estimates of regression coefficients is described. ZMM models are mostly based on Poisson or negative binomial type 2 distribution (NB2). In this work, author has extended the theory to ZIM models generally based on any discrete distributions of exponential type. There is described a construction of MLE of regression coefficients of theese models, too. Just few of present works are interested in ZIM models based on negative binomial type 1 distribution (NB1). This distribution is not of exponential type therefore a common method of MLE construction in ZIM models cannot be used here. In this work provides modification of this method using quasilikelihood method. There are two simulation studies concluding the work. 1
