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The illustration of the law of large numbers by simulations
Chabičovský, Martin ; Kříž, Oldřich (referee) ; Michálek, Jaroslav (advisor)
Stochastic convergence, law of large numbers and central limit theorem is an important part of probability theory, which is often used in mathematical statistics. The aim of this work is to describe this theory and demonstrate it with examples and graphical simulation. In addition simulation of stochastic convergence, law of large numbers and central limit theorem for some discrete and continuous probability distribution the work contains several interesting simulations for example simulation of Galton's box, Buffon's needle problem and Bertrand's paradox. To create a graphic simulation were used programming language matlab.
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Small sample asymptotics
Tomasy, Tomáš ; Sabolová, Radka (advisor) ; Omelka, Marek (referee)
In this thesis we study the small sample asymptotics. We introduce the saddlepoint approximation which is important to approximate the density of estimator there. To derive this method we need some basic knowledge from probability and statistics, for example the central limit theorem and the M- estimators. They are presented in the first chapter. In practical part of this work we apply the theoretical background on the given M-estimators and selected distribution. We also apply the central limit theorem on our estimators and compare it with small sample asymptotics. At the end we show and summarize the calculated results.
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Central and Non-Central Limit Theorems
Kiška, Boris ; Čoupek, Petr (advisor) ; Beneš, Viktor (referee)
V této práci zkoumáme centrální limitní věty (CLT) a jejich různé varianty. Zpočátku je uvedena CLT pro nezávislé a stejně rozdělené náhodné veličiny. Dále studujeme případ nezávislých a nestejně rozdělených náhodných veličin, kde porovnáme různé verze a různé podmínky, za kterých CLT platí. Tyto klasické výsledky jsou prezentovány spolu s několika protipříklady, které porušují předpoklady CLT různými způsoby. V této práci je také uvažován případ závislých náhodných veličin. Zejména CLT pro a-mixující náhodné posloupnosti je dána společně s Rosenblattovým protipříkladem, který zahrnuje limitní ne- Gaussovské rozdělení, které se nyní nazývá Rosenblattovo rozdělení.
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Three proofs of a limit theorem
Marcinčín, Martin ; Štěpán, Josef (advisor) ; Beneš, Viktor (referee)
We show three diferent proofs of the central limit theorem using elementary methods. The central limit theorem with the Feller - Lindeberg condition is proven using a convergence of charakteristic functions and Fejer theorem about uniform convergence of trigonometric polynoms on a bounded interval. The second proof is based on the fact that convergence in distribution is equivalent to convergence of means of functions with all derivatives bounded. The central limit theorem for sums of independent random variables with all moments finite is shown using convergence of all moments and determinacy of normal distribution by its moments.
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Edgeworth expansion
Dzurilla, Matúš ; Omelka, Marek (advisor) ; Nagy, Stanislav (referee)
This thesis is focused around Edgeworth's expansion for approximation of distribution for parameter estimation. Aim of the thesis is to introduce term Edgeworth's expansion, its assumptions and terminology associated with it. Afterwards demonstrate process of deducting first term of Edgeworth's 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 Edgeworth's expansion.
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Chaotic random variables in applied probability
Večeřa, Jakub ; Beneš, Viktor (advisor) ; Reitzner, Matthias (referee) ; Pawlas, Zbyněk (referee)
This thesis deals with modeling of particle processes. In the first part we ex- amine Gibbs facet process on a bounded window with discrete orientation distri- bution and we derive central limit theorem (CLT) for U-statistics of facet process with increasing intensity. We calculate all asymptotic joint moments for interac- tion U-statistics and use the method of moments for deriving the CLT. Moreover we present an alternative proof which makes use of the CLT for U-statistics of a Poisson facet process. In the second part we model planar segment processes given by a density with respect to the Poisson process. Parametric models involve reference distributions of directions and/or lengths of segments. Statistical methods are presented which first estimate scalar parameters by known approaches and then the reference distribution is estimated non-parametrically. We also introduce the Takacs-Fiksel estimate and demonstrate the use of estimators in a simulation study and also using data from actin fibres from stem cells images. In the third part we study a stationary Gibbs particle process with determin- istically bounded particles on Euclidean space defined in terms of a finite range potential and an activity parameter. For small activity parameters, we prove the CLT for certain statistics of this...
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Continuity correction
Štěpán, Marek ; Omelka, Marek (advisor) ; Maciak, Matúš (referee)
For an approximation of discrete random variable, which is the sum of n inde- pendent, identically distributed discrete random variables, we can use the central limit theorem. However, it turns out that we can refine this approximation by applying continuity correction. This term is explained in the thesis, and it is illustrated several ways how the continuity correction can be derived. There is also a numerical comparison of the approximation error for the binomial distribu- tion approximation by the normal distribution with the correction for continuity and approximation without the correction. There are also described confidence intervals and χ2 test of independence in contingency tables in which continu- ity correction are used. On simulations for various parameters, we will test the properties of these intervals (true confidence level and length) and tests (actual significance level and power).
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Central and Non-Central Limit Theorems
Kiška, Boris ; Čoupek, Petr (advisor) ; Beneš, Viktor (referee)
V této práci zkoumáme centrální limitní věty (CLT) a jejich různé varianty. Zpočátku je uvedena CLT pro nezávislé a stejně rozdělené náhodné veličiny. Dále studujeme případ nezávislých a nestejně rozdělených náhodných veličin, kde porovnáme různé verze a různé podmínky, za kterých CLT platí. Tyto klasické výsledky jsou prezentovány spolu s několika protipříklady, které porušují předpoklady CLT různými způsoby. V této práci je také uvažován případ závislých náhodných veličin. Zejména CLT pro a-mixující náhodné posloupnosti je dána společně s Rosenblattovým protipříkladem, který zahrnuje limitní ne- Gaussovské rozdělení, které se nyní nazývá Rosenblattovo rozdělení.
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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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Selected problems of random walks
Pavčová, Eva ; Hlubinka, Daniel (advisor) ; Pawlas, Zbyněk (referee)
Title: Selected problems of random walks Author: Eva Pavčová Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Daniel Hlubinka, Ph.D., Department of Probability and Mathematical Statistics Abstract: This thesis deals with simple random walks and solutions of theoretical selected problems. We define the path which can be interpreted as the realization of a random walk. We bring forward examples of paths with illustrations and basic properties such as ballot theorem and reflection principle. Random walk is defined and also the probability of its is brought forward. Our attention is concentrated on the main lemma. We derive from it other interesting assertions such as arcsin law. The aim of this thesis is to solve the selected problems using theoretical knowledge. The problems are concerned with probabilities and numbers of paths with certain restrictions. The specific problem of positive paths proves geometrically the equality of numbers of two types of paths. Specially, we are interested in the proof of reformulation of main lemma. Keywords: path, reflection principle, main lemma, arcsin law
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