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Statistic Characteristic Function and its Usage for Digital Signal Processing
Mžourek, Zdeněk ; Mekyska, Jiří (referee) ; Smékal, Zdeněk (advisor)
Aim of this thesis is provide basic information about characteristic function used in statistic and compare its properties with the Fourier transform used in engineering applications. First part of this thesis is theoretical, there are discussed basic concepts, their properties and mutual relations. The second part is devoted to some possible applications, for example normality testing of data or utilization of the characteristic function in independent component analysis. The first chapter describes the introduction to probability theory for the unification of terminology and mentioned concepts will be used to demonstrate the interesting properties of characteristic function. The second chapter describes the Fourier transform, definition of characteristic function and their comparison. The second part of this text is devoted to applications the empirical characteristic function is analyzed as an estimate of the characteristic function of examined data. As an example of application is describe a simple test of normality. The last part deals with more advanced applications of characteristic function for methods such as independent component analysis.
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Statistical analysis of compound distributions
Konečný, Zdeněk ; Druckmüller, Miloslav (referee) ; Michálek, Jaroslav (advisor)
The probability distribution of a random variable created by summing a random number of the independent and identically distributed random variables is called a compound probability distribution. In this work is described a compound distribution as well as a calculation of its characteristics. Especially, the thesis is focused on studying a special case of compound distribution where each addend has the log-normal distribution and their number has the negative binomial distribution. Here are also described some approaches to estimate the parameters of LN and NB distribution. Further, the impact of these estimates on the final compound distribution is analyzed.
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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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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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Sums of exponential random variables
Michl, Marek ; Seidler, Jan (advisor) ; Maslowski, Bohdan (referee)
Sums of exponential random variables are often found in applied mathematics. Their densities are known and are well documented in engineering articles. However, these articles usually lack detailed deductions. The purpose of this thesis is to give rigorous deductions of explicit formulas for densities of sums of independent exponential random variables, which are known. The thesis covers several cases depending on whether the variables have the same distribution or not. Furthermore, the thesis gives a summary of basic characteristics of exponential distribution and the relation between sums of identically distributed exponential random variables and a random variable with gamma distribution. Based on this relation the density of the sum of gamma random variables with the same intestity is given. Powered by TCPDF (www.tcpdf.org)
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Maximum likelihood estimators and their approximations
Tyuleneva, Anastasia ; Omelčenko, Vadim (advisor) ; Zvára, Karel (referee)
Title: Maximum likelihood estimators and their approximations Author: Anastasia Tyuleneva Department: Department of Probability and Mathematical Statistics Supervisor: Mgr. Vadym Omelchenko Abstract: Maximum likelihood estimators method is one of the most effective and accurate methods that was used for estimation distributions and parameters. In this work we will find out the pros and cons of this method and will compare it with other estimation models. In the theoretical part we will review important theorems and definitions for creating common solution algorithms and for processing the real data. In the practical part we will use the MLE on the case study distributions for estimating the unknown parameters. In the final part we will apply this method on the real price data of EEX A. G, Germani. Also we will compare this method with other typical methods of estimation distributions and parameters and chose the best distribution. All tests and estimators will be provided by Mathematica software. Keywords: parametr estimates, Maximum Likelihood estimators, MLE, Stable distribution, Characteristic function, Pearson's chi-squared test, Rao-Crámer. .
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Statistical analysis of compound distributions
Konečný, Zdeněk ; Druckmüller, Miloslav (referee) ; Michálek, Jaroslav (advisor)
The probability distribution of a random variable created by summing a random number of the independent and identically distributed random variables is called a compound probability distribution. In this work is described a compound distribution as well as a calculation of its characteristics. Especially, the thesis is focused on studying a special case of compound distribution where each addend has the log-normal distribution and their number has the negative binomial distribution. Here are also described some approaches to estimate the parameters of LN and NB distribution. Further, the impact of these estimates on the final compound distribution is analyzed.
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Statistic Characteristic Function and its Usage for Digital Signal Processing
Mžourek, Zdeněk ; Mekyska, Jiří (referee) ; Smékal, Zdeněk (advisor)
Aim of this thesis is provide basic information about characteristic function used in statistic and compare its properties with the Fourier transform used in engineering applications. First part of this thesis is theoretical, there are discussed basic concepts, their properties and mutual relations. The second part is devoted to some possible applications, for example normality testing of data or utilization of the characteristic function in independent component analysis. The first chapter describes the introduction to probability theory for the unification of terminology and mentioned concepts will be used to demonstrate the interesting properties of characteristic function. The second chapter describes the Fourier transform, definition of characteristic function and their comparison. The second part of this text is devoted to applications the empirical characteristic function is analyzed as an estimate of the characteristic function of examined data. As an example of application is describe a simple test of normality. The last part deals with more advanced applications of characteristic function for methods such as independent component analysis.
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