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Berry-Esseen inequality
Hezoučký, Martin ; Pawlas, Zbyněk (advisor) ; Maslowski, Bohdan (referee)
In this submitted bachelor thesis we deal with the convergence in dis- tribution rate estimates of the standardized mean of independent identically dis- tributed random variables to standard normal distribution. In the introductory part we present applied probability distributions and state necessary mathemati- cal theorems - primarily the central limit theorem and Berry-Esseen's theorem. In the application part, we deal with specific probability distributions and deter- mine for them all parameters affecting their convergence rate. In particular, we focus on estimates of the constant C from Berry-Esseen's inequality for these se- lected distributions. Subsequently we compare the final estimates of the constant C for these probability distributions and deal with its coherence with the theo- retical estimates of this constant. 1
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Discrete scan statistics
Láf, Adam ; Pawlas, Zbyněk (advisor) ; Beneš, Viktor (referee)
The discrete scan statistic is defined as the maximum of moving sums of a given number of consecutive observations in a sequence of i.i.d. integer valued random variables. This thesis introduces various ways to approximate the distri- bution of the discrete scan statistic. These approximations are evaluated based on enumerations in specific cases. The main focus is on random variables with Bernoulli distribution, the only case where exact results for the distribution of the discrete scan statistic are available. Some connections with well-known problems as the birthday problem and the longest success run in a sequence of Bernoulli trials are also discussed. 1
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Simulated annealing
Seitl, Filip ; Beneš, Viktor (advisor) ; Pawlas, Zbyněk (referee)
Simulated annealing is a probabilistic optimization algorithm which is used for approximating the global extremes of a function at a large state space. We construct a homogeneous Markov chain, whose stationary distribution is depen- dent on the temperature parameter (this distribution is called the Boltzmann distribution), on this space. With declining the parameter this distribution fo- cuses on the states minimizing the function. The algorithm, on which it can be viewed as a non-homogeneous Markov chain, we use to solve the hard-core model and the graph bisection. We will also deal with the convergence of the algorithm, too rapidly decreasing sequence of the parameters can result in stucking in a lo- cal extreme of the function, therefore some requirements on this sequence will be determined. 1
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Spatial econometrics
Nývltová, Veronika ; Pawlas, Zbyněk (advisor) ; Kopa, Miloš (referee)
This thesis is devoted to the models that are suitable for modelling spatial data. For this purpose, random fields with finite index set are used. Based on the neighbourhood relationship a spatial weight matrix is introduced which describes spatial dependencies. A recognition and testing of spatial dependence is mentioned and it is applied for macroeconomic indicators in the Czech Republic. Spatial models originated from generalization of usual time series models are subsequently combined with linear regression models. The parameter estimators are derived for selected models by three different methods. These methods are ordinary least squares, maximum likelihood and method of moments. Theoretical asymptotic results are supplemented by a simulation study that examines the performance of estimators for finite sample size. Finally, a short illustration on real data is demonstrated. Powered by TCPDF (www.tcpdf.org)
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Urn models with stochastic replacements
Kochaniková, Petra ; Pawlas, Zbyněk (advisor) ; Dvořák, Jiří (referee)
The thesis purpose is to discuss urn models where the probability of success at any trial depends upon the number of previous successes. Such a scheme allows us to estimate the number of HIV cases among intravenous drug users. The coef- ficients in known probability generating function will be derived for the number of new infectives generated in both homogenous and inhomogenous population. The expectations and variances of the number of new infectives are also derived for both cases. These derived values will be verified for some fixed number of infectives and susceptibles by simulations. In the end of this thesis the studied model will be applied on a practical example where the effect of vaccination will be studied. 1
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Random marked sets
Kráľová, Veronika ; Beneš, Viktor (advisor) ; Pawlas, Zbyněk (referee)
In this thesis, two models of marked point processes are investigated. One of the marks have a continuous distribution on a compact Riemannian manifold. The von Mises distribution and its properties are studied. Metropolis-Hastings algorithm of Markov chain Monte Carlo method is used for the simulation of Gibbs segment process. Takacs-Fiksel estimator and its modified version are examined. A kernel density estimator and entropy estimator are proposed and applied to simulated and real data. Powered by TCPDF (www.tcpdf.org)
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Models of marked point processes
Héda, Ivan ; Pawlas, Zbyněk (advisor) ; Dvořák, Jiří (referee)
Title: Models of Marked Point Processes Author: Ivan Héda Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Zbyněk Pawlas, Ph.D. Abstract: In the first part of the thesis, we present necessary theoretical basics as well as the definition of functional characteristics used for examination of marked point patterns. Second part is dedicated to review some known marking strategies. The core of the thesis lays in the study of intensity-marked point processes. General formula for the characteristics is proven for this marking strategy and general class of the models with analytically computable characteristics is introduced. This class generalizes some known models. Theoretical results are used for real data analysis in the last part of the thesis. Keywords: marked point process, marked log-Gaussian Cox process, intensity-marked point process 1
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MCMC methods for financial time series
Tritová, Hana ; Pawlas, Zbyněk (advisor) ; Komárek, Arnošt (referee)
This thesis focuses on estimating parameters of appropriate model for daily returns using the Markov Chain Monte Carlo method (MCMC) and Bayesian statistics. We describe MCMC methods, such as Gibbs sampling and Metropolis- Hastings algorithm and their basic properties. After that, we introduce different financial models. Particularly we focus on the lognormal autoregressive model. Later we theoretically apply Gibbs sampling to lognormal autoregressive model using principles of Bayesian statistics. Afterwards, we analyze procedu- res, that we used in simulations of posterior distribution using Gibbs sampling. Finally, we present processed output of both simulated and real data analysis.
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Existence and uniqueness of the distribution of a random measure given by finite dimensional projections
Jurčo, Adam ; Rataj, Jan (advisor) ; Pawlas, Zbyněk (referee)
Title: Existence and uniqueness of the distribution of a random measure given by finite dimensional projections Author: Adam Jurčo Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Jan Rataj, CSc., Department of Probability and Mathe- matical Statistics Abstract: This thesis deals with the existence and uniqueness of the distribu- tion of a random measure given a system of finite-dimensional distributions. A random measure can be interpreted as a particular system of random variables. Conversely, we will want to know what conditions would allow a system of random variables to be extended to a random measure and if this extension is unique. We will start with a consistent system of finite-dimensional distributions and use Daniell-Kolmogorov theorem to find the necessary and sufficient conditions for the existence of such extension. A counterexample will be included to show that it is not possible to use this theory for random signed measures. Keywords: Random measure, point process, finite-dimensional distributions. 1
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Random closed sets and particle processes
Stroganov, Vladimír ; Rataj, Jan (advisor) ; Pawlas, Zbyněk (referee)
In this thesis we are concerned with representation of random closed sets in Rd with values concentrated on a space UX of locally finite unions of sets from a given class X ⊂ F. We examine existence of their repre- sentations with particle processes on the same space X, which keep invariance to rigid motions, which the initial random set was invariant to. We discuss existence of such representations for selected practically applicable spaces X: we go through the known results for convex sets and introduce new proofs for cases of sets with positive reach and for smooth k-dimensional submanifolds. Beside that we present series of general results related to representation of random UX sets. 1
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