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Skew normal distribution
Helebrand, František ; Hudecová, Šárka (advisor) ; Antoch, Jaromír (referee)
Title: Skew normal distribution Author: František Helebrand Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Šárka Hudecová, Ph.D., Department of probability and mathematical statistics Abstract: In this paper, the skewed normal probability distribution is studied. First, the density is proposed and the basic properties of this distribution are proved. Then the thesis deals with the moment and cumulative generating functions. These functions are used in deriving the mean, variance and skewness of the skew normal distribution. In the third chapter, two parameter estimators are proposed and their properties are derived. Finally, these estimators are empirically investigated in a simulation study and on real data. Keywords: normal distribution, skewness, point estimators 1
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Estimation of the standard deviation based on the mean absolute deviation
Lipavská, Kateřina ; Hudecová, Šárka (advisor) ; Maciak, Matúš (referee)
This thesis deals with the average absolute deviation and its use for the estimation of the standard deviation. First, the important notions are defined, in particular the mean absolute deviation and the average absolute deviation. Relationships between the standard deviation and the mean absolute deviation are derived for four probability dis- tributions, namely for the normal, exponential, Laplace distribution and for a mixture of two normal distributions. The asymptotic distribution of the average absolute deviation is derived. Subsequently, the estimation of the standard deviation via the average ab- solute deviation is considered and its asymptotic distribution is shown. Detailed results are provided for the normal and Laplace distribution. Some of the theoretical results are illustrated by simulations. In addition, the estimator of the standard deviation construc- ted using the average absolute deviation is compared to the sample standard deviation in a simulation study. 1
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Modeling categorical time series
Jarina, Vesna ; Zichová, Jitka (advisor) ; Hudecová, Šárka (referee)
This bachelor thesis is primary focused on introducing models for categorical time series of nominal and ordinal type, based on the theory of generalized linear models (GLM). The theoretical part also deals with the problem of parameter estimation using the partial likelihood method. Finally, the practical part intro- duces an application of both models on simulated data and investigates the rate of convergence of the maximum partial likelihood estimators (MPLE). 1
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Goodness-of-fit tests based on the empirical probability generating function
Mečiarová, Kristína ; Hudecová, Šárka (advisor) ; Hlávka, Zdeněk (referee)
A choice of a proper parametric model for a given data is often a crucial question. This thesis deals with goodness-of-fit tests for Poisson distribution. The tests are based on a comparison of the empirical probability generating function with the theoretical Poisson generating function, or its parametric estimator. Tests for both simple and composite hypothesis are introduced. The comparison of the functions is made at one point and at more points and asymptotic distribution of the particular test statistics is derived. A simulation study is conducted in order to examine the choice of the number of points and their values that leads to the most powerful tests. The presented methodology is illustrated on real data analysis of monthly polio incidence and in analysis of chromosome aberrations as result of radiation exposure. 1
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Verification of linear mixed model assumptions
Krnáč, Ľuboš ; Kulich, Michal (advisor) ; Hudecová, Šárka (referee)
1 AbstraktEN The diploma thesis deals with linear mixed effects models. In the first chap- ter, we discuss parameter estimation and hypothesis testing in the linear mixed effects models. The second chapter is dedicated to graphical diagnostics. We look at the suitable diagnostic plots for residuals and random effects estimates. It is closely described, how the violations of assumptions affect the diagnostic plots. In the third chapter we have consequences of the violations of assumptions on the parameter estimates and results of hypothesis testing for fixed effects. 1
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Binomial autoregressive model
Hledík, Jakub ; Hudecová, Šárka (advisor) ; Prášková, Zuzana (referee)
Binomial AR(1) process is a model for integer-valued time series with a fi- nite range and discrete time. It has the binomial marginal distribution and the AR(1)-like autocorrelation structure. This thesis deals with deriving some ba- sic properties of this process, methods of parameter estimation and goodness of fit testing. Three methods of parameter estimation are presented: Yule-Walker, the conditional least squares and the maximum likelihood method together with proofs of their asymptotical properties. Next, the goodness of fit testing is pre- sented. At first, two known methods based on the marginal distribution and the autocorrelation function are summarized. Then our own method is added, based on the probability generating function. Several simulations are provided to show the properties of all tests. The application of this model is illustrated on a real dataset. 1
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Index of dispersion for discrete distributions
Semjonov, Valerij ; Hudecová, Šárka (advisor) ; Zichová, Jitka (referee)
This thesis deals with the index of dispersion for discrete distributions. In the first chapter, we define the sample index of dispersion and describe it's basic properties , specifically for the Poisson distribution. An asymptotic distribution of the sample index of dispersion will be derived for the Poisson and some other distributions. In the second chapter, we describe the index of dispersion test and determine it's approximate power against some specific alternatives. The third chapter is dedicated to a simulation study in which statistical properties of the test are investigated. Empirical estimation of the power of the test will be compared with the analytical results obtained in the second chapter.
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Probability forecast in exponential smoothing models
Viskupová, Barbora ; Hudecová, Šárka (advisor) ; Cipra, Tomáš (referee)
This thesis deals with the use of statistical state space models of exponential smooth- ing for estimating the conditional probability distribution of future values of time series. This knowledge allows calculation of interval predictions, not only point forecasts. Meth- ods of exponential smoothing are described and set into the context of state space models. Analytical and simulation methods used in the calculation of interval predictions are presented, in particular simulations based on assumption of normality, bootstrap method or estimated parametric model. The methods are applied to simulated as well as real data and their results are compared. 1
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Bivariate negative binomial distributions
Šír, David ; Hudecová, Šárka (advisor) ; Hlávka, Zdeněk (referee)
The thesis summarizes basic properties of the negative binomial distribution, including estimations of unknown parameters which are derived with the help of the method of moments and the maximum likelihood method. The main part of the thesis describes the bivariate negative binomial distribution. Basic properties of the studied distribution are derived. For instance marginal distribution, distribution of the sum of elements and conditional distribution are negative binomial. The unknown parameters are estimated using the methods of moments and maximum likelihood method. The consistency and asymptotic normality of these estimators are proved. The final sample behaviour of the estimators is investigated in a small simulation study. The described bivariate distribution is applied to real traffic accidents data set from the Czech Republic. 1
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Robust estimation of autocorrelation function
Lain, Michal ; Hudecová, Šárka (advisor) ; Hlávka, Zdeněk (referee)
The autocorrelation function is a basic tool for time series analysis. The clas- sical estimation is very sensitive to outliers and can lead to misleading results. This thesis deals with robust estimations of the autocorrelation function, which is more resistant to the outliers than the classical estimation. There are presen- ted following approaches: leaving out the outliers from the data, replacement the average with the median, data transformation, the estimation of another coeffici- ent, robust estimation of the partial autocorrelation function or linear regression. The thesis describes the applicability of the presented methods, their advantages and disadvantages and necessary assumptions. All the approaches are compared in simulation study and applied to real financial data. 1
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