
Variance stabilizing transformations
Kuželová, Noemi ; Omelka, Marek (advisor) ; Komárek, Arnošt (referee)
Abstract. We often examine data whose sample mean converges to a normal distribution, but the variance generally depends on an unknown parameter. To get rid of this dependence, we can sometimes use the socalled variancestabilizing transformation method. Firstly, this thesis explains the method in detail and finds a general procedure to find suitable transformations. Then it will focus on data from Poisson and binomial distributions with unknown parameters. For these data, it finds transformations that stabilize (asymptotic) variance, and compares them with the "improved"transforms from the article Anscombe (1948). Most of the thesis is devoted to the shape of these transformations. Finally, we show in the Poisson distribution simulation that it is really appropriate to use this method and compare the derived transformation with its Anscombe version.


Continuity correction
Štěpán, Marek ; Omelka, Marek (advisor) ; Maciak, Matúš (referee)
For an approximation of discrete random variable, which is the sum of n inde pendent, identically distributed discrete random variables, we can use the central limit theorem. However, it turns out that we can refine this approximation by applying continuity correction. This term is explained in the thesis, and it is illustrated several ways how the continuity correction can be derived. There is also a numerical comparison of the approximation error for the binomial distribu tion approximation by the normal distribution with the correction for continuity and approximation without the correction. There are also described confidence intervals and χ2 test of independence in contingency tables in which continu ity correction are used. On simulations for various parameters, we will test the properties of these intervals (true confidence level and length) and tests (actual significance level and power).


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


The method of reweighting (calibration) in survey sampling
Michálková, Anna ; Omelka, Marek (advisor) ; Antoch, Jaromír (referee)
In this thesis, we study reweighting when estimating totals in survey sampling. The purpose of reweighting is to adjust the structure of the sample in order to comply with the structure of the population (with respect to given auxiliary variables). We sum up some known results for methods of the traditional desinbased approach, more attention is given to the modelbased approach. We generalize known asymptotic results in the modelbased theory to a wider class of weighted estimators. Further, we propose a consistent estimator of asymptotic variance, which takes into consideration weights used in estimator of the total. This is in contrast to usually recommended variance estimators derived from the designbased approach. Moreover, the estimator is robust againts particular model misspecifications. In a simulation study, we investigate how the proposed estimator behaves in comparison with variance estimators which are usually recommended in the literature or used in practice. 1


Edgeworth expansion
Dzurilla, Matúš ; Omelka, Marek (advisor) ; Nagy, Stanislav (referee)
This thesis is focused around Edgeworths expansion for aproximation of distribution for parameter estimation. Aim of the thesis is to introduce term Edgeworths expansion, its assumptions and terminology associeted with it. Afterwords demonstrate process of deducting first term of Edgeworths expansion. In the end demonstrate this deduction on examples and compare it with different approximations (mainly central limit theorem), and show strong and weak points of Edgeworths expansion.


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.
