
Statistical models for detection of differential item functioning
Hladká, Adéla ; Martinková, Patrícia (advisor) ; Wiberg, Marie (referee) ; Hlávka, Zdeněk (referee)
This thesis focuses on topic of Differential Item Functioning (DIF), a phenomenon that can arise in various contexts of educational, psychological, or healthrelated multi item measurements. We discuss several statistical methods and models to detect DIF among dichotomous, ordinal, and nominal items. In the first part, generalized logistic regression models for DIF detection among di chotomous items are introduced, which account for possibility of guessing and/or inat tention. Techniques for estimation of item parameters are presented, including a newly proposed algorithm based on a parametric link function. Two simulation studies are presented. The first compares the generalized logistic regression models to other widely used DIF detection methods. The second illustrates differences between the techniques to estimate item parameters. Implementation of the models into the R software and its difNLR package is illustrated. In the second part, generalized logistic regression models for DIF detection among polytomous items are discussed. Cumulative logit, adjacent category logit, and nominal models are introduced together with the maximum likelihood method to estimate item parameters and with examples of implementation in the difNLR package. The third part deals with a nonparametric comparison of regression...


Goodnessoffit tests for Pareto distribution
Oganesyan, Robert ; Hlávka, Zdeněk (advisor) ; Zichová, Jitka (referee)
Pareto model is widely used in insurance and reinsurance, as well as in finance (capital management, stock returns), and in modelling of natural disasters. As a motivation, we introduce a case study drawn from the pricing of reinsurance excess of loss contracts. In the first chapter of the thesis, we define four types of Pareto distributions and explore such properties as moments, parameter estimation and characterizations. In the follow ing chapter, we examine the theory of unbiased estimation in order to better understand the construction of goodnessoffit tests based on characterizations. Subsequently, we introduce an overview of some goodnessoffit tests. The final chapter focuses on simu lating the type I error rates and power of these tests, as well as conducting a comparative analysis. Finally, we return to the motivational example and complete the calculation of the price for the reinsurance coverage. 1


Tests of independence for functional data
Horská, Šárka ; Hlávka, Zdeněk (advisor) ; Hušková, Marie (referee)
This thesis deals with tests of serial independence for functional time series. The first part of the thesis introduces the issue of serial independence in time series of random vari ables. The second part focuses on tests of serial independence for functional observations. It examines a test based on autocorrelation and, in particular, a test whose test statistic is derived directly from the definition of independence. This test is modified to a test with a weaker alternative of subdependence. The thesis concludes with a comparison of these three tests, which is based on a simulation. 1


Statistical inference in varying coefficient models
Cichrová, Michaela ; Maciak, Matúš (advisor) ; Hlávka, Zdeněk (referee)
In this master thesis we study varying coefficient models, which is a class of models that allow the coefficients to be smooth functions of some effectmodifying variable. We introduce the models in a broader context and then focus only on longitudinal settings. We consider two splinebased methods to estimate the coefficient functions, the poly nomial spline approach and the smoothing spline approach. For the polynomial spline approach, we derive its asymptotic properties, which we use to construct asymptotic confidence intervals and bands. We assess the performance of the confidence bands in a small simulation study, considering two slight modifications of the construction. 1


Goodnessoffit tests for Poisson distribution based on zero index
Váňová, Julie ; Hudecová, Šárka (advisor) ; Hlávka, Zdeněk (referee)
This bachelor thesis deals with goodnessoffit tests for Poisson distribution that are based on socalled zero index. In the first part, Poisson zero index is defined and some of its basic properties are discussed. Further, asymptotic distribution of zero indexes is derived and it is used to construct asymptotic goodnessoffit tests. Particular examples of zero indexes and related tests are included. In the following part, other types of goodnessoffit tests for Poisson distribution are briefly described, in particular χ2 tests and tests based on index of dispersion. All mentioned methods are then compared in a simulation study. 1


Multicollinearity
Dřizgová, Lucie ; Zvára, Karel (advisor) ; Hlávka, Zdeněk (referee)
In our work, we explored multicollinearity problem from a complex point of view  from diagnostic methods to the solving of the problems which are caused by the multicollinearity. We compared the Least Squares method with some alternative methods  Principal Component Regression, Partial Least Squares Regression and Ridge Regression on the theoretical basis. In the last section, we demonstrated all methods on practical example computed in the program R.


Tests of independence in contingency tables
Pavlík, Lukáš ; Maciak, Matúš (advisor) ; Hlávka, Zdeněk (referee)
In this thesis we investigate various methods for testing independence in twoway contingency tables. The methods are explained, their advantages and drawbacks are dis cussed, and we also illustrate the methods on an example. Further, we compare the tests on simulated data using R statistic programming language. Based on simulation results we try to decide which test is the best choice for a situation. In particular, we investi gate a new method, USP test, which is based on the theory of so called Ustatistics. We therefore describe these, too. It is shown that USP test performs much better than other tests in particular cases, but fails in some others. These cases are specified and guidelines are made about when the test is advantageous to use and when it is not. 1


Models for equating in cognitive tests
Vařejková, Michaela ; Martinková, Patrícia (advisor) ; Hlávka, Zdeněk (referee)
This thesis focuses on statistical methods for equating cognitive tests, which is the process of transforming scores from multiple test versions to ensure their comparability. Divided into three chapters, the theoretical part of the thesis addresses different approaches to test equating. The first chapter presents tra ditional equating methods, the second explores kernel equating methods, while the third covers equating methods using Item Response Theory models. The concluding part of the thesis showcases an empirical study demonstrating the application of equating methods on a real dataset. This dataset contains respon ses to two versions of a math test taken by fourthgrade students in the Czech Republic as part of the 2015 TIMSS international survey. 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


Mean estimation in normal distribution
Kaliská, Andrea ; Zichová, Jitka (advisor) ; Hlávka, Zdeněk (referee)
The bachelor's thesis deals with estimators of the nonconstant mean value in a specific probability model, which was taken from the article Estimating the Current Mean of a Normal Distribution which is Subjected to Changes in Time. Firstly, we describe this model in detail and we add proofs. We consider two estimators: the minimum variance linear unbiased estimator and the Bayes estimator in cases with at most one change, which is taken directly from the article. The thesis is concluded with a simulation study, focusing on the comparison of those estimators. 1
