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The method of re-weighting (calibration) in survey sampling
Michálková, Anna ; Omelka, Marek (advisor) ; Antoch, Jaromír (referee)
In this thesis, we study re-weighting when estimating totals in survey sampling. The purpose of re-weighting 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 desin-based approach, more attention is given to the model-based approach. We generalize known asymptotic results in the model-based 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 design-based 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
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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.
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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 non-inferiority 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 duo-trio, 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
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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 well-known Cochran-Mantel-Haenszel test based on a hypergeometric distribution can be found. This chapter also includes a description of Breslow-Day test of homogeneous association. A numerical simulation of chosen tests is performed eventually. 1
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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
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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
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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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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. K-sample 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.
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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 non-negative. 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 non-negativity. Finally, simulations are conducted to evaluate the finite sample performances of the mentioned estimators.
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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
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