
Testing equivalence and noninferiority
Rychterová, Nela ; Antoch, Jaromír (advisor) ; Omelka, Marek (referee)
This master thesis deals with topics related to the task whether customers are able to recognize a difference between products. First, testing of equivalence and noninferiority is discussed in detail. It is an important tool when verifying that two products are equivalent or that a new product is not substantially worse than a current product. Afterwards, Thurstone's approach is introduced as a way to evaluate the impact of a stimulus on human senses. Subsequently, using the previous chapters, there is a detailed discussion dealing with three standards wi dely used in practice in the case when someone needs to apply sensory evaluation to verify whether customers are able to recognize a difference between products. In particular, these are duotrio, triangle and paired comparison tests. There is a thorough explanation of their statistical base and the tests are compared accor ding to their power. Furthermore, an approach based on the Thurstone's theory is introduced as an alternative to the standard methods. Moreover, this thesis introduces Saaty's approach to the estimation of a priority vector, which is a useful tool to compare, to order or to choose the best one from n objects. We also introduce another approach to estimation of a priority vector which is based on Saaty's idea. 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


Calibration Estimators in Survey Sampling
Klička, Petr ; Omelka, Marek (advisor) ; Nagy, Stanislav (referee)
V této práci se zabýváme odhady populačního úhrnu s využitím pomoc ných informací. V práci je popsán obecný regresní odhad a předpoklady, za kterých je splněna asymptotická normalita tohoto odhadu. Dále jsou zde po psány kalibrační odhady a zmínka o jejich asymptotické ekvivalenci s obec ným regresním odhadem. Odvozené závěry aplikujeme na data z RADIO PROJEKTu a porovnáme je s výsledky získanými společnostmi, které tento projekt realizovali. Na závěr pomocí simulací porovnáme skutečné pravdě podobnosti pokrytí interval· spolehlivosti pro populační úhrn spočítané na základě teorie uvedené v této práci a na základě metod společností realizu jících RADIOPROJEKT. 1


Computerized adaptive testing in knowledge assessment
Tělupil, Dominik ; Martinková, Patrícia (advisor) ; Omelka, Marek (referee)
In this thesis, we describe and analyze computerized adaptive tests (CAT), the class of psychometrics tests in which items are selected based on the actual estimate of respondent's ability. We focus on the tests based on di chotomic IRT (item response theory) models. We present critera for item selection, methods for ability estimation and termination criteria, as well as methods for exposure rate control and content balancing. In the analytical part, the effect of CAT settings on the average length of the test and on absoulute bias of ability estimates is investigated using linear regression mo dels. We provide post hoc analysis of real data coming from real admission test with unknown true values of abilities, as well as simulation study based on the simulated answers of respondents with known true values of ability. In the last chapter of the thesis we investigate the possibilities of analysing adaptive tests in R software and of creating a real CAT. 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.


Maximum likelihood theory for not i.i.d. observations
Kielkowská, Eva ; Omelka, Marek (advisor) ; Pešta, Michal (referee)
Maximum likelihood approach for independent but not identically distributed observations is studied. In the first part of the thesis, conditions for consistency and asymptotic normality of the maximum likelihood estimates for this case are stated. Uniform integrability has a major role in proving the desired properties. Ksample problem serves as an example for using the described method. The second part is focused on estimates obtained by minimizing convex functions. Convexity is a key for showing the consistency and asymptotic normality of the estimates in this case. The results can be used for maximum likelihood when observations with logconcave densities are involved. Finally, normal linear model, logistic regression and Poisson regression examples are provided to present the application of the method.


Nonparametric regression estimators
Měsíček, Martin ; Hlávka, Zdeněk (advisor) ; Omelka, Marek (referee)
This thesis is focused on local polynomial smoothers of the conditional vari ance function in a heteroscedastic nonparametric regression model. Both mean and variance functions are assumed to be smooth, but neither is assumed to be in a parametric family. The basic idea is to apply a local linear regression to squa red residuals. This method, as we have shown, has high minimax efficiency and it is fully adaptive to the unknown conditional mean function. However, the local linear estimator may give negative values in finite samples which makes variance estimation impossible. Hence Xu and Phillips proposed a new variance estimator that is asymptotically equivalent to the local linear estimator for interior points but is guaranteed to be nonnegative. We also established asymptotic results of both estimators for boundary points and proved better asymptotic behavior of the local linear estimator. That motivated us to propose a modification of the local li near estimator that guarantees nonnegativity. Finally, simulations are conducted to evaluate the finite sample performances of the mentioned estimators.


Tests for Paired Categorical Data
Míchal, Petr ; Komárek, Arnošt (advisor) ; Omelka, Marek (referee)
In this paper we deal with paired categorical data. We will test marginal ho mogeneity and symmetry of corresponding probability table. At first, we describe multinomial distribution and contingency tables. In the next section, we deal with dichotomic paired categorical data, we derive McNemar's test and describe test for small sample sizes. Further, we state tests for general paired categorical data, Stuart's and Bhapkar's test are described. We then state test derived by Bowker, which is used for testing symmetry of probability table. In the last section, we show simulations of McNemar's test in software R. 1


Statistical inference in multivariate distributions based on copula models
Kika, Vojtěch ; Omelka, Marek (advisor) ; Hlubinka, Daniel (referee)
Diploma thesis abstract Thesis title: Statistical inference in multivariate distributions based on copula models Author: Vojtěch Kika This diploma thesis aims for statistical inference in copula based models. Ba sics of copula theory are described, followed by methods for statistical inference. These are divided into three main groups. First of them are parametric methods for copula parameter estimation which assume fully parametric structure, thus for both joint and marginal distributions. The second group consists of semi parametric methods for copula parameter estimation which, unlike parametric methods, do not require parametric structure for marginal distributions. The last group describes goodnessoffit tests used for testing the hypothesis that consi dered copula belongs to some specific copula family. The thesis is accompanied by a simulation study that investigates the dependence of the observed coverage of the asymptotic confidence intervals for copula parameter on the sample size. Pseudolikelihood method was chosen for the simulation study since it is one of the most popular semiparametric methods. It is shown that sample size of 50 seems to be sufficient for the observed coverage to be close to the theoretical one. For Frank and GumbelHougaard copula families even sample size of 30 gives us...


Archimedean copulas
Vedyushenko, Anna ; Pešta, Michal (advisor) ; Omelka, Marek (referee)
The thesis deals with Archimedean copulas which are very popular nowadays due to easy construction and their appealing properties. At first it introduces the general definition of a copula and also shows its fundamental properties. After that the definition and the basic properties of an Archimedean copula are discussed. The paper also describes some of the commonly used families of Archi medean copulas. Then several methods of parameter estimation for Archimedean copulas are shown. Finally, we make a study of two real datasets where the distri bution of the data is estimated based on the procedures described in the thesis. 1
