
Binomial autoregressive model
Hledík, Jakub ; Hudecová, Šárka (advisor) ; Prášková, Zuzana (referee)
Binomial AR(1) process is a model for integervalued 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: YuleWalker, 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


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.


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


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


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


Linear regression model with autocorrelated residuals
Kostka, Ján ; Zichová, Jitka (advisor) ; Hudecová, Šárka (referee)
The aim of this bachelor thesis is to introduce the algorithm for analysis of the linear regression model with autocorrelated residuals, which is applicable to time series data. For residuals, we assume the ARMA model, eventually ARIMA model, which enlarges the possibilities of application. The analysis of such regression models includes detection of autocorrelation and related tests, detection of stationarity and related unit root test, followed by model identification for residuals and maximum likelihood estimation of identified regression model.


Change detection in RCA models
Biolek, Jiří ; Prášková, Zuzana (advisor) ; Hudecová, Šárka (referee)
The thesis describes Random Coefficient Autoregressive time series mo dels (RCA models). In first chapter we introduce different types of estimati ons for coefficients of RCA model. Main part is in second chapter, where we describe detection changes procedures for all methods mentioned in chapter one, here the thesis expands the current theory about change detection of wei ghted least square method and functional estimation. In last chapter we sum marize results of simulation study. 1


Markov binomial model
Šuléřová, Natálie ; Hudecová, Šárka (advisor) ; Dvořák, Jiří (referee)
In this thesis we study the Markov chain binomial model, which generalizes the standard binomial distribution. Instead of the sum of independent random vari ables, we consider the sum of random variables that form a stationary Markov chain. The goal of this thesis is to describe this model along with its proper ties, such as the expected value, variance and probability generating function. A part of this thesis is dedicated to estimating parameters of this model using the method of moments and the maximum likelihood estimation. The accuracy of the methods is compared in a simulation study and obtained results are dis cussed. The presented model is then applied on a real dataset based on rate of alcoholimpaired car accidents.


Basic stochastic epidemic models
Strachoňová, Karla ; Hudecová, Šárka (advisor) ; Kulich, Michal (referee)
This thesis deals with two basic models which are used for epidemic model ling in closed populations, namely Greenwood and ReedFrost models. At first, knowledge which a reader needs to have about Markov chains and random varia bles is summarized. Then the two models are described by modelling the number of susceptible and infectious individuals, as well as the duration and size of the epidemic. All of these approaches to modelling an epidemic are then illustrated on examples. Finally, the maximum likelihood method of the probability of infection is described and illustrated on real data in the last chapter, where the obtained results are discussed as well. 1


Poisson autoregression
Böhmová, Karolína ; Hudecová, Šárka (advisor) ; Hlubinka, Daniel (referee)
This thesis deals with INGARCH models for a count time series. Main emphasis is placed on a linear INARCH model. Its properties are derived. Several methods of estimation are introduced  maximum likelihood method, least squares method and its modifications  and later compared in a simulation study. Main properties and maximum likelihood estimation for INGARCH(1,1) model are stated. Higher order linear INGARCH models and nonlinear INGARCH models are discussed briefly. An application of the presented models on time series of car accidents is given.
