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Love-Young Inequality and Its Consequences
Sýkora, Adam ; Čoupek, Petr (advisor) ; Hlubinka, Daniel (referee)
This thesis is focused on proving the Love-Young inequality and clarifying the manner in which it relates to a fractional Brownian motion. To begin with, several estimates alongside the concept of p-variation of a func- tion are presented. The connection between functions of finite p-variation and regulated functions is then highlighted and used to prove the aforementioned Love-Young inequality. Deficiency of the pathwise approach to stochastic in- tegration is recognised and later discussed amongst the properties of fractional Brownian motions. This constitutes the main application of the featured theory which is the integration with respect to irregular functions. 1
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Weighted Data Depth and Depth Based Discrimination
Vencálek, Ondřej ; Hlubinka, Daniel (advisor) ; Anděl, Jiří (referee) ; Malý, Marek (referee)
The concept of data depth provides a powerful nonparametric tool for multivariate data analysis. We propose a generalization of the well-known halfspace depth called weighted data depth. The weighted data depth is not affine invariant in general, but it has some useful properties as possible nonconvex central areas. We further discuss application of data depth methodology to solve discrimination problem. Several classifiers based on data depth are reviewed and one new classifier is proposed. The new classifier is a modification of k-nearest- neighbour classifier. Classifiers are compared in a short simulation study. Advantage gained from use of the weighted data depth for discrimination purposes is shown.
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Statistical inference for random processes
Kvitkovičová, Andrea ; Hlubinka, Daniel (advisor) ; Štěpán, Josef (referee)
The thesis deals with testing hypotheses about the parameters of the Wiener process with a constant drift rate and instantaneous variance. The tests are based on the first time, when the process reaches a pre-specified boundary point. We consider a process with a non-negative drift rate, and we observe hitting a positive point. We focus on tests about the drift rate, in particular about the absence of any drift. We first study several basic properties of the Wiener process and its connection with the Wiener process with a drift. Using these, we derive distributional properties of the first hitting time. We also describe selected hypotheses testing techniques in the setting of exponential families. We construct uniformly most powerful unbiased tests of one parameter in the presence of a nuisance parameter. Further, we construct uniformly most powerful tests of hypotheses about the drift rate, while the variance is known, and we study this situation in more detail. Finally, we construct asymptotic simultaneous tests of both parameters based on the R'enyi divergences.
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Dynamic fare model
Kislinger, Jan ; Lachout, Petr (advisor) ; Hlubinka, Daniel (referee)
The problem of creating dynamic fare model consists of two tasks - estimating demand for train tickets and multistage optimization of price of fare. We introduce a model of inhomogeneous Markov process for the process of selling the tickets in this thesis. Because of the complexity of the state space the optimization problem needs to be solved using simulation methods. The solution was implemented in R language for single-stage and two-stage problems. Before this application we summarize the theory of inhomogeneous Markov process with special attention to process with separable inhomogeneity. Then we propose methods for estimating the intensity using maximum likelihood theory. We also describe and compare two algorithms for simulated optimization. Powered by TCPDF (www.tcpdf.org)
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Deterministic and Stochastic Epidemic Models
Staněk, Jakub ; Štěpán, Josef (advisor) ; Hlubinka, Daniel (referee) ; Dohnal, Gejza (referee)
Kermack-McKendrick model and its version with vaccination are presented. First, we introduce a model with vaccination and then a numerical study that includes comparison of di erent vaccination strategies and searching for optimal vaccination strategy is presented. We proceed to introduce a stochastic model with migration and consequently we suggest its generalization and prove the existence and uniqueness of a solution to the stochastic di erential equation (henceforth SDE) describing this model. Three stochastic versions of Kermack-McKendrick model with vaccination are suggested and compared. A procedure of nding the optimal vaccination strategy is presented. We also prove the theorem on the existence and uniqueness of a solution to the SDE that drives a model with multiple pathogens. Finally, the stochastic di erential equation describing the general model is presented. We study properties of a solution to this SDE and present sufficient conditions for the existence of a solution that is absorbed by the natural barrier of the model.
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Comparison of exact and asymptotic tests
Rusnáková, Viktória ; Hlubinka, Daniel (advisor) ; Došlá, Šárka (referee)
Nazev prace: Porovnani pfesnych a asymptotickych testu Autor: Viktoria Rusnakova Katedra: Katcdra pravdepodobnosti a matematicke statistiky Vedouci bakalafske prace: Doc. RNDr. Daniel Hlubinka, Ph.D. e-mail vedouciho: hlubinka(Q)karlin.mff.cuni.cz Abstrakt: Predlozena praca sa zaobera testami o parametroch binomickeho. Poiaso novho a cxponencialneho rozdclenia. Ciol'om prace je porovnat! presne znahodneiie testy a testy asymptoticke, zalozcne na centralnej limitnej veto a zalozcne na metode maximalnej vierohodnosti. V prvej, teoretickej casti su uvedene potrebne vety a tvrdcnia, na zaklado ktorych su zostrojenc priklady pro dane rozdclenia. Druha, prakticka cast' je venovana pocitacovym simulaciarn obojstrannych testov na hlacline vyznamnosti 5 %. Vysledky siraulacii su uvedene na prilozenom CD. Klfcova slova: kriticka funkcia, znahodneny test, asymptoticky test Title: Comparison of exact and asymptotic tests Author: Viktoria Rusnakova Department: Department of Probability and Mathematical Statistics Supervisor: Doc. RNDr. Daniel Hlubinka, Ph.D. Supervisor's e-mail address: hlubinkaykarhn.mff.cuin.cz Abstract: The presented work deals with tests about parameters of binomial, Poisson and exponencial distribution. The aim of this work is to compare exact randomized tests and asymptotic tests based on central...
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