
Confidence regions in nonlinear regression
Marcinko, Tomáš ; Zvára, Karel (advisor) ; Komárek, Arnošt (referee)
The aim of this thesis is a comprehensive description of the properties of a nonlinear least squares estimator for a nonlinear regression model with normally distributed errors and thorough development of various methods for constructing confidence regions and confidence intervals for the parameters of the nonlinear model. Due to the fact that, unlike the case of linear models, there is no easy way to construct an exact confidence region for the parameters, most of these methods are only approximate. A short simulation study comparing observed coverage of various confidence regions and confidence intervals for models with different curvatures and sample sizes is also included. In case of negligible intrinsic curvature the use of likelihoodratio confidence regions seems the most appropriate.


Analysis of crossover clinical trials in the presence of baseline measurements
Helebrand, František ; Kulich, Michal (advisor) ; Komárek, Arnošt (referee)
This thesis aims to provide a comprehensive overview of methods for estimating treat ment effects in crossover designs. It examines approaches that use baseline measurements to estimate the treatment effect, as well as alternative methods that do not use baseline measurements at all. It also proposes new approaches to estimating treatment effects and introduces robust procedures to ensure that biases caused by residual treatment effects from the previous period are within acceptable limits. The theoretical properties of the methods are investigated in a simulation study. Furthermore, a comparison of different methods is performed in cases where a theoretical comparison is not possible. 1


Estimation of latent distribution for ordinal data
Hržič, Viktor ; Hudecová, Šárka (advisor) ; Komárek, Arnošt (referee)
The main goal of the bachelor thesis is to introduce problematics of ordinal data together with estimations of latent density distribution based on ordinal data. The esti mations obtained using ordinal data are compared to the ones that are more common and used on daily basis. The reader will be introduced to the maximum likelihood method that is used in the development of each estimate. One chapter is dedicated to alterna tive approach using Bernstein polynomials. When we take all benefits into account such as easier collecting the data in ordinal form or minimization of possible errors committed by respondent, we obtained very valuable and promising results. 1


Classification based on mixture models
Janečková, Lucie ; Komárek, Arnošt (advisor) ; Maciak, Matúš (referee)
This thesis deals with classification based on mixture models, mainly on models finite normal. At first, there are introduced basic definitions and characteristics of finite mix tures. Afterwards there is described the maximum likelihood method and her obstacles in context of finite mixtures, which we are using for unknown parameters estimation. Then there is described EM algorithm, that is used to obtain the maximum likelihood estimator and there are calculated the formulae for one iteration of EM algorithm. In the last part there is shown, how can finite normal mixtures be used for classification. 1


Dynamic prediction in survival analysis
Mečiarová, Kristína ; Komárek, Arnošt (advisor) ; Pešta, Michal (referee)
Often the motivation behind building a statistical model is to provide prediction for an outcome of interest. In the context of survival analysis it is important to distingu ish between two types of timevarying covariates and take into careful consideration the appropriate type of analysis. Joint model for longitudinal and timetoevent data, in con trast to standard Cox model, enables to account for continuous change of the covariate over time in the survival model. In this thesis two examples of joint models are presen ted, the shared randomeffect model and the joint latent class model. Bayesian estimation of the model parameters and summary of methodology for dynamic prediction of indi vidual survival probability is provided for the first one of the aforementioned types of models. Application of the theoretical knowledge is illustrated in the analysis of the data on primary biliary cirrhosis. The impact of number of patients, number of longitudinal measurements and percent of censoring on the quality of prediction and estimates of the model parameters is examined in the simulation study. 1


Tolerance limits
Bedřich, Marek ; Omelka, Marek (advisor) ; Komárek, Arnošt (referee)
This bachelor's thesis deals with tolerance intervals, a statistical tool used to quan tify the uncertainty of statistical predictions. The introductory part of the text briefly recalls confidence intervals. The thesis then focuses on prediction intervals, which are an intermediate step between confidence intervals and tolerance intervals. Specifically, the prediction interval for normal distribution and nonparametric prediction interval are analyzed. The main part of the thesis then deals with tolerance intervals  the definition, construction of both parametric and nonparametric tolerance intervals, convergence, and actual coverage of the derived intervals. In the final part, an example of the practical application of this tool is presented. 1


Modelbased Clustering of Multivariate Longitudinal Data of a Mixed Type
Vávra, Jan ; Komárek, Arnošt (advisor) ; FrühwirthSchnatter, Sylvia (referee) ; Hlávka, Zdeněk (referee)
Modelbased Clustering of Multivariate Longitudinal Data of a Mixed Type Jan Vávra October 3, 2022 Abstract In many nowadays studies, the data are collected repeatedly on the same units over a certain period of time. Moreover, such longitudinal data are composed of numeric values, count variables, binary indicators, ordered or nominal categories. A few variants of statistical model capa ble of modelling such often highly correlated data jointly are introduced. On top of that, a methodology of modelbased clustering is adapted to such models to discover hidden heterogeneity within the data by dividing units into clusters of specific characteristics. Bayesian approach is taken, generative model is proposed and MCMC methodology is developed for estimation. A simulation study verifying the estimation properties is con ducted. The methodology is applied to real datasets such as medical data on patients suffering from primary biliary cholangitis (PBC) or econom ical dataset consisting of thousands of Czech households followed since 2005 (EUSILC database). 1


Modelbased Clustering of Multivariate Longitudinal Data of a Mixed Type
Vávra, Jan ; Komárek, Arnošt (advisor)
Modelbased Clustering of Multivariate Longitudinal Data of a Mixed Type Jan Vávra October 3, 2022 Abstract In many nowadays studies, the data are collected repeatedly on the same units over a certain period of time. Moreover, such longitudinal data are composed of numeric values, count variables, binary indicators, ordered or nominal categories. A few variants of statistical model capa ble of modelling such often highly correlated data jointly are introduced. On top of that, a methodology of modelbased clustering is adapted to such models to discover hidden heterogeneity within the data by dividing units into clusters of specific characteristics. Bayesian approach is taken, generative model is proposed and MCMC methodology is developed for estimation. A simulation study verifying the estimation properties is con ducted. The methodology is applied to real datasets such as medical data on patients suffering from primary biliary cholangitis (PBC) or econom ical dataset consisting of thousands of Czech households followed since 2005 (EUSILC database). 1


Modeling transition intensities of a nonhomogenous Markov chain via the Cox model
Jandl, Vojtěch ; Kulich, Michal (advisor) ; Komárek, Arnošt (referee)
We study the extension of methods from the classical twostate survival analysis to the multistate setting. Such models are applicable in a variety of fields in situations, for which the classical case does not suffice due to the fact that omission of some states is not possible. First of all, we explore one sample methods, in particular the extensions of the well known NelsonAalen and KaplanMeier estimators. Then, we deal with regression models for transition intensities, which include a generalisation of the Cox model, and the LinYing additive model, for which we derive the asymptotic properties of the estimator of regression parameters. Lastly, to illustrate the practicality of presented methods, we propose an experiment that could help a website understand the behaviour of its members. A small simulation study is also a part of the last chapter in order to demonstrate the asymptotic properties of the underlying model empirically. 1


Classification based on mixture models
Janečková, Lucie ; Komárek, Arnošt (advisor) ; Maciak, Matúš (referee)
This thesis deals with classification based on mixture models, mainly on models finite normal. At first, there are introduced basic definitions and characteristics of finite mix tures. Afterwards there is described the maximum likelihood method and her obstacles in context of finite mixtures, which we are using for unknown parameters estimation. Then there is described EM algorithm, that is used to obtain the maximum likelihood estimator and there are calculated the formulae for one iteration of EM algorithm. In the last part there is shown, how can finite normal mixtures be used for classification. 1
