 

Deconvolution
Dibák, Miroslav ; Hurt, Jan (referee) ; Hlávka, Zdeněk (advisor)
The convolution has a big signification in mathematical statistics. In the opening chapter, we define basic terms used in the thesis and we introduce the convolution and basic relations related to this term. In the second chapter, we attend to kernel estimators, mainly the kernel density estimator and the kernel charakteristic function estimator. In the third chapter, we attend to the deconvolution and we summarize the basic theoretical properties of the deconvolution estimator. In the last chapter of this thesis we present a possible application in medicine. The properties of the proposed estimator are investigated in a small simulation study.


Analysis of change from baseline to postintervention value
Pacáková, Andrea ; Hlávka, Zdeněk (referee) ; Kulich, Michal (advisor)
The aim of the present work is to compare three di erent estimators of a treatment e ect in clinical randomized studies. The purpose of these studies is to compare the change of a distribution of certain variable between two attendances. Mentioned estimators were developed from the assumption of validity of some model. In this work we gather properties of the estimators when each of all given models is valid. We deal with the consistency of the estimators and with their asymptotic distributions and then we compare the estimators on the basis of their asymptotic variances. In the most of cases is possible to make the comparison in general. In the case when it is not possible, we show a few particular examples. Eventually, we accomplish the simulation study, which certi es theoretical conclusions and extends pieces of knowledge in the cases when it was not possible to make theoretical computation in general.


Multicollinearity
Dřizgová, Lucie ; Hlávka, Zdeněk (referee) ; Zvára, Karel (advisor)
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.

 

Analysis of Biosensoric Data
Timková, Jana ; Antoch, Jaromír (referee) ; Hlávka, Zdeněk (advisor)
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Testing of composite hypotheses in regression models with small sample size
Simerská, Olga ; Hlávka, Zdeněk (referee) ; Kulich, Michal (advisor)
The thesis, through a simulation study, examines the behaviour of asymptotic tests for testing hypotheses that several coefficients in logistic models are zero. Likelihood ratio, Wald's, and Rao's tests are considered. The necessary theory is formulated to derive the form of the statistics of asymptotic tests for testing composite hypotheses in logistic regression. Based on the numerical treatment of simulated data, the levels of significance of these tests are investigated, with critical values of the chisquared distribution. The powers of the tests are then compared, modified empirically so that all tests reject the null hypotesis at the 5% level. The main focus is on the dependence of these values on the sample size and parameter settings.


Isotonic Regression in Sobolev Spaces
Pešta, Michal ; Dostál, Petr (referee) ; Hlávka, Zdeněk (advisor)
We propose a class of nonparametric estimators for the regression models based on least squares over the sets of sufficiently smooth functions. Least squares permit the imposition of additional constraintisotoniaon nonparametric regression estimation and testing of this constraint. The estimation takes place over the balls of functions which are elements of a suitable Sobolev spacespecial types of Hilbert spaces that facilitate calculation of the least squares projection. The Hilbertness is allowing us to take projections and hence to decompose spaces into mutually orthogonal complements. Then we transform the problem of searching for the best fitting function in an infinite dimensional space into a finite dimensional optimization problem. We prove that the balls of functions in Sobolev space are bounded and have bounded higher order derivatives. It permits us to estimate over rich set of functions with sufficiently low metric entropy and apply Laws of Large Numbers and Central Limit Theorems.
