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Extreme Value Distributions with Applications
Fusek, Michal ; Skalská,, Hana (referee) ; Karpíšek, Zdeněk (referee) ; Michálek, Jaroslav (advisor)
The thesis is focused on extreme value distributions and their applications. Firstly, basics of the extreme value theory for one-dimensional observations are summarized. Using the limit theorem for distribution of maximum, three extreme value distributions (Gumbel, Fréchet, Weibull) are introduced and their domains of attraction are described. Two models for parametric functions estimation based on the generalized extreme value distribution (block maxima model) and the generalized Pareto distribution (threshold model) are introduced. Parameters estimates of these distributions are derived using the method of maximum likelihood and the probability weighted moment method. Described methods are used for analysis of the rainfall data in the Brno Region. Further attention is paid to Gumbel class of distributions, which is frequently used in practice. Methods for statistical inference of multiply left-censored samples from exponential and Weibull distribution considering the type I censoring are developed and subsequently used in the analysis of synthetic musk compounds concentrations. The last part of the thesis deals with the extreme value theory for two-dimensional observations. Demonstrational software for the extreme value distributions was developed as a part of this thesis.
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Bayesian and Maximum Likelihood Nonparametric Estimation in Monotone Aalen Model
Timková, Jana ; Volf, Petr (advisor) ; Kraus, David (referee) ; Komárek, Arnošt (referee)
This work is devoted to seeking methods for analysis of survival data with the Aalen model under special circumstances. We supposed, that all regression functions and all covariates of the observed individuals were nonnegative and we named this class of models monotone Aalen models. To find estimators of the unknown regres- sion functions we considered three maximum likelihood based approaches, namely the nonparametric maximum likelihood method, the Bayesian analysis using Beta processes as the priors for the unknown cumulative regression functions and the Bayesian analysis using a correlated prior approach, where the regression functions were supposed to be jump processes with a martingale structure.
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EM algorithm
Vacula, Ondřej ; Komárek, Arnošt (advisor) ; Antoch, Jaromír (referee)
This paper discusses the EM algorithm. This algorithm is used, for example, to calculate maximum likelihood estimate of unknown parameter. The algorithm is based on repeated calculations of certain expected value and maximizing specific function. We begin with parameter estimation problem, describe the maximum likelihood method and concept of incomplete data. Then we formulate the EM algorithm and its properties. In the next chapter we apply this knowledge to three selected statistical problems. At first we examine standard mixture model, then the linear mixed model and finally we analyze censored data. Powered by TCPDF (www.tcpdf.org)
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Maximum likelihood methods; selected problems
Chlubnová, Tereza ; Hlubinka, Daniel (advisor) ; Hlávka, Zdeněk (referee)
Maximum likelihood estimation is one of statistical methods for estimating an unknown parameter. It is often used because of a simple calculation of the estimator and also for characteristics of this estimator, which the method provides under some conditions. In the thesis we prove a consistence of the estimator under conditions of regularity and uniqueness of the root of the likelihood equation. If we add other assumptions we show its asymptotic normality and we expand this result from the one-dimensional parameter to the multi-dimensional parameter. The main result of the thesis lies in exercises, in which we cannot express the maximum likelihood estimator in general, but we can show its existence, uniqueness and asymptotic normality. Moreover we demonstrate the utilization of asymptotic normality of the estimator for asymptotic hypothesis tests and confidence intervals of the parameter. Powered by TCPDF (www.tcpdf.org)
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Expectation-Maximization Algorithm
Vichr, Jaroslav ; Pešta, Michal (advisor) ; Zvára, Karel (referee)
EM (Expectation-Maximization) algorithm is an iterative method for finding maximum likelihood estimates in cases, when either complete data include missing values or assuming the existence of additional unobserved data points can lead to more simple formulation of the model. Each of its iterations consists of two parts. During the E step (expectation) we calculate the expected value of the log-likelihood function of the complete data, with respect to the observed data and the current estimate of the parameter. The M step (maximization) then finds new estimate, which will maximize the function obtained in the previous step and which will be used in the next iteration in step E. EM algorithm has important use in e.g. price and manage risk of the portfolio.
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Bayesian and Maximum Likelihood Nonparametric Estimation in Monotone Aalen Model
Timková, Jana
This work is devoted to seeking methods for analysis of survival data with the Aalen model under special circumstances. We supposed, that all regres- sion functions and all covariates of the observed individuals were nonnegative and we named this class of models monotone Aalen models. To find estimators of the unknown regression functions we considered three maximum likelihood based approaches, namely the nonparametric maximum likelihood method, the Bayesian analysis using Beta processes as the priors for the unknown cumulative regression functions and the Bayesian analysis using a correlated prior approach, where the regression functions were supposed to be jump processes with a martingale structure. Powered by TCPDF (www.tcpdf.org)
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Score tests in contingency tables
Jex, Martin ; Omelka, Marek (advisor) ; Kulich, Michal (referee)
The thesis deals with testing of hypotheses in multinomial distribution. It utilizes two approaches, Pearson's approach known as the of goodness of fit test and the approach stemming from theory of maximum likelihood. The thesis presents derivations of tests based on maximum likelihood. Both approaches are used on the multinomial distribution and for both cases with and without nuisance parameters. The links between both approaches are presented as well. Furthermore both approaches are illustrated on real data to facilitate better understanding of the discussed problems. 1
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Incomplete Poisson samples
Zeman, Ondřej ; Dvořák, Jiří (advisor) ; Hlubinka, Daniel (referee)
The topic of my bachelor thesis is studying truncated Poisson sample which is a part of a sample from Poisson distribution, where zero observations are missing. The main goal is estimating the size of the original sample and the parameter λ of the Poisson distribution. In the first chapter I mainly focus on deriving three types of estimators of these parameters and I describe their basic properties. Second chapter contains simulations where the estimators from the first chapter are compared based on the estimates of relative bias and relative mean square error. Eventually in the third chapter I focus on the asymptotic properties of derived estimators with emphasis on consistency of estimators. 1
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Maximum likelihood methods; selected problems
Chlubnová, Tereza ; Hlubinka, Daniel (advisor) ; Hlávka, Zdeněk (referee)
Maximum likelihood estimation is one of statistical methods for estimating an unknown parameter. It is often used because of a simple calculation of the estimator and also for characteristics of this estimator, which the method provides under some conditions. In the thesis we prove a consistence of the estimator under conditions of regularity and uniqueness of the root of the likelihood equation. If we add other assumptions we show its asymptotic normality and we expand this result from the one-dimensional parameter to the multi-dimensional parameter. The main result of the thesis lies in exercises, in which we cannot express the maximum likelihood estimator in general, but we can show its existence, uniqueness and asymptotic normality. Moreover we demonstrate the utilization of asymptotic normality of the estimator for asymptotic hypothesis tests and confidence intervals of the parameter. Powered by TCPDF (www.tcpdf.org)
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Bayesian and Maximum Likelihood Nonparametric Estimation in Monotone Aalen Model
Timková, Jana ; Volf, Petr (advisor) ; Kraus, David (referee) ; Komárek, Arnošt (referee)
This work is devoted to seeking methods for analysis of survival data with the Aalen model under special circumstances. We supposed, that all regression functions and all covariates of the observed individuals were nonnegative and we named this class of models monotone Aalen models. To find estimators of the unknown regres- sion functions we considered three maximum likelihood based approaches, namely the nonparametric maximum likelihood method, the Bayesian analysis using Beta processes as the priors for the unknown cumulative regression functions and the Bayesian analysis using a correlated prior approach, where the regression functions were supposed to be jump processes with a martingale structure.
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