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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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Bayesian modeling of market price using autoregression model
Šindelář, Jan ; Kárný, Miroslav (advisor) ; Pawlas, Zbyněk (referee) ; Šmíd, Martin (referee)
1 Bayesian modeling of market price using autoregression model 1Šindelář Jan Department: Department of Probability and Mathematical Statistics Supervisor: Ing. Miroslav Kárný, DrSc. Abstract: In the thesis we present a novel solution of Bayesian filtering in autoregression model with Laplace distributed innovations. Estimation of regression models with lep- tokurtically distributed innovations has been studied before in a Bayesian framework [2], [1]. Compared to previously conducted studies, the method described in this article leads to an exact solution for density specifying the posterior distribution of parameters. Such a solution was previously known only for a very limited class of innovation distributions. In the text an algorithm leading to an effective solution of the problem is also proposed. The algorithm is slower than the one for the classical setup, but due to increasing com- putational power and stronger support of parallel computing, it can be executed in a reasonable time for models, where the number of parameters isn't very high. Keywords: Bayesian, Autoregression, Optimal Trading, Time Series References [1] P. Congdon. Bayesian statistical modelling. Wiley, 2006. [2] A. Zellner. Bayesian and Non-Bayesian analysis of the regression model with multivari- ate Student-t error term. Journal...
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Set-indexed stochastic processes
Schenk, Martin ; Pawlas, Zbyněk (advisor) ; Rataj, Jan (referee)
This thesis deals with the problem of estimating the joint probability distribution of a marked process' parameters from a censored data. First, a Nelson-Aalen estimator of the cumulative hazard rate for one-dimensional case is constructed. This estimator is then smoothed by using a kernel function estimator. Then, a Kaplan-Meier estimator of the survival function is brought in. Further, a theory of set-indexed random processes is built up to be a base for the construction of a generalized Nelson-Aalen estimator of the cumulative hazard rate, which is then again smoothed. For a special case, a generalized Kaplan-Meier estimator of the multidimensional survival function is constructed. The application of the mentioned generalized estimators is shown on a particular case. These estimators are then used on simulated data.
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Coupon's collector problem
Nývltová, Veronika ; Pawlas, Zbyněk (advisor) ; Bártek, Jan (referee)
In the presented work we are looking for the answer to the question how many purchases must be made for obtaining a collection of cards. As a collection we understand all types of cards, which are packaged with products or we consider a collection of chosen types of these cards. First it is assumed that all cards are uniformly distributed. The number of required purchases is random and we derive its mean value, variance and probability distribution. We study limit behaviour when the number of types of cards is going to infinity. We are looking for the answer to the same question in the case of collecting several collections of cards at the same time. These collections could be complete or incomplete. In the case that cards are not uniformly distributed we describe mean value and variance of the number of purchases necessary for acquiring several collections of cards.
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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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Analysis of number lotteries
Jedličková, Veronika ; Pawlas, Zbyněk (advisor) ; Lachout, Petr (referee)
This bachelor thesis focuses on most well-known lotteries on the Czech market, in particular Sportka and Loto. Thesis observes many aspects influencing progress of these games. Winnning prices and lottery participant's expectations are examined. Total sum of these winnings is influenced by the amount of money in jackpot. Therefore, jackpot sum modelling and period between wins is taken into account. Moreover, expected period between two jackpot hits, distribution of drawn numbers and probability of drawing the same winning sequence is examined.
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