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Tests for the Poisson distribution
Trusina, Filip ; Pawlas, Zbyněk (advisor) ; Nagy, Stanislav (referee)
In this work we deal with the question whether a sequence of independent identically distributed random variables comes from the Poisson distribution. For this task we present two different approaches and couple of tests for each appro- ach. The first approach is based on the asymptotic approximation of distribution of test statistics. The second approach uses generation of test samples. Based on simulations done by us, we discuss the power of individual tests and their advantages and disadvantages. 1
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Essential problems of random walks
Michálek, Matěj ; Hlubinka, Daniel (advisor) ; Pawlas, Zbyněk (referee)
In this paper, we cover some essential problems of (simple) random walks in one, two and three dimensions. At the begining, we work only in one dimension. We find the probability of a position on a line at particular time. Then we study returns to origin and examine if return to origin is certain. Also, we look into a theorem called the arc sine law. Furthermore, we generalise some of those problems into two and three dimensions. We investigate a probability of a position in time and space and returns to origin. 1
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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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Second-order characteristics of point processes
Gupta, Archit ; Pawlas, Zbyněk (advisor) ; Prokešová, Michaela (referee)
In this thesis we examine estimation of the K-function which is an important second-order characteristic in the theory of spatial point processes. Besides Ripley's K-function based on a spherical structuring element we also work with the multiparameter K-function where the struc- turing element is rectangular. We consider the Poisson point process model, which is the fundamental model for complete spatial randomness. We de- rive expressions for both bias and variance of the estimators. The primary goal of this thesis is the study of different edge correction methods that are available for the K-function. Using simulations we also study a few variance approximations proposed in the literature and compare them with empirical variances. 1
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Stochastic Differential Equations with Gaussian Noise
Janák, Josef ; Maslowski, Bohdan (advisor) ; Duncan, Tyrone E. (referee) ; Pawlas, Zbyněk (referee)
Title: Stochastic Differential Equations with Gaussian Noise Author: Josef Janák Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Bohdan Maslowski, DrSc., Department of Probability and Mathematical Statistics Abstract: Stochastic partial differential equations of second order with two un- known parameters are studied. The strongly continuous semigroup (S(t), t ≥ 0) for the hyperbolic system driven by Brownian motion is found as well as the formula for the covariance operator of the invariant measure Q (a,b) ∞ . Based on ergodicity, two suitable families of minimum contrast estimators are introduced and their strong consistency and asymptotic normality are proved. Moreover, another concept of estimation using "observation window" is studied, which leads to more families of strongly consistent estimators. Their properties and special cases are descibed as well as their asymptotic normality. The results are applied to the stochastic wave equation perturbed by Brownian noise and illustrated by several numerical simula- tions. Keywords: Stochastic hyperbolic equation, Ornstein-Uhlenbeck process, invariant measure, paramater estimation, strong consistency, asymptotic normality.
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Nonparametric tests of independence
Kmeťková, Diana ; Pawlas, Zbyněk (advisor) ; Hlubinka, Daniel (referee)
The main objective of this thesis is the presentation regarding the problem of testing independence between two random variables in the nonparametric model of continuous cumulative distribution functions. Firstly, the reader is informed with basic notions from the theory of independence and rank tests. Afterwards, few of the most common methods for testing independence are introduced. In the beginning, the test based on Pearson's correlation coefficient is mentioned as a representative for parametric tests, then we continue with nonparametric tests, such as test based on Spearman's, Kendall's and distance correlation coefficient. We focus in better detail on Hoeffding's test of independence, which results to be consistent against all alternatives in the model of continuous cumulative distribution functions. In the end, we compare and evaluate presented methods for testing independence using simulations in R environment.
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Multivariate point processes and their application on neurophysiological data
Bakošová, Katarína ; Pawlas, Zbyněk (advisor) ; Prokešová, Michaela (referee)
This thesis examines a multivariate point process in time with focus on a mu- tual relations of its marginal point processes. The first chapter acquaints the re- ader with the theoretical background of multivariate point processes and their properties, especially the higher-order cumulant-correlation measures. Later on, several models of multivariate point processes with different dependence structu- res are characterized, such as the random superposition model, a Poisson depen- dent superposition point process, a jitter Poisson dependent superposition point process orrenewal processes models. Simulations of each of them are provided. Furthermore, two statistical methods for higher-order correlations are presented; the cumulant based inference of higher-order correlations, and the extended til- ling coefficient. Finally, the introduced methods are applied not only on the data from simulations, but also on the real, simultaneously recorded nerve cells spike train data. The results are discussed. 1
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Non-homogeneous Poisson process - estimation and simulation
Vedyushenko, Anna ; Pešta, Michal (advisor) ; Pawlas, Zbyněk (referee)
This thesis covers non-homogeneous Poisson processes along with estimation of the intensity (rate) function and some selected simulation methods. In Chapter 1 the main properties of a non-homogeneous Poisson process are summarized. The main focus of Chapter 2 is the general maximum likelihood estimation procedure adjusted to a non-homogeneous Poisson process, together with some recommen- dations about calculation of the initial estimates of the intensity function param- eters. In Chapter 3 some general simulation methods as well as the methods designed specially for log linear and log quadratic rate functions are discussed. Chapter 4 contains the application of the described estimation and simulation methods on real data from non-life insurance. Furthermore, the considered sim- ulation methods are compared with respect to their time efficiency and accuracy of the simulations. 1
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Buffon needle problem and its generalizations
Hledík, Jakub ; Pawlas, Zbyněk (advisor) ; Prokešová, Michaela (referee)
This thesis contains detailed derivation of results of several generalizations of the Buffon needle problem. Next to the original problem we study grids composed of rectangles, known as Buffon-Laplace needle problem, then grids composed of parallelograms, triangles or hexagons. The application of this problem is briefly shown on the estimation of π, additional references are mentioned. We provide a proof of the theorem computing the area of a polygon, if the Cartesian coordi- nates of its vertices are known. Finally, we show how to solve grids composed of several different shapes, this is demonstrated on the grid composed of a regular hexagon and a diamond. 1
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