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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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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 Reed-Frost 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
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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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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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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, non-trivial 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 Clopper-Pearson, Wald, Wilson and logit. 1
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Parameter estimation of gamma distribution
Zahrádková, Petra ; Kulich, Michal (advisor) ; Hlávka, Zdeněk (referee)
It is well-known 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 closed-form 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 closed-forms enable bias-corrections to these estimators. 1
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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.
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Post-selection 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 post-selection inference for the lasso estimates where we explain why the classical inference is not suitable. Three post-selection 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
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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 Kolmogorov-Smirnov 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)
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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
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