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Parameter estimating in time series models
Kostárová, Aneta ; Zichová, Jitka (advisor) ; Prášková, Zuzana (referee)
This bachelor thesis deals with some methods of parameter estimating in linear time series models. The most used approach in software products is the maximum likelihood estimation. The theoretical part explains the parameter estimation of the ARMA model by conditional and unconditional maximum likelihood estimation and demonstrates both methods for lower order models. The practical part examines and describes the imple- mentation of parameter estimating in Mathematica and R software. The comparison of the quality of the estimates calculated by various procedures of the chosen software is included. Finally, the acquired findings is used in a simulation study. 1
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Sparse contingency tables
Smítková, Viktorie ; Hlávka, Zdeněk (advisor) ; Prášková, Zuzana (referee)
This bachelor thesis deals with the issue of independence testing in sparse contingency tables. It defines contingency tables and describes their creation and properties. It pre- sents the most commonly used independence tests and proposes tests suitable for the problem of independence testing in sparse contingency tables. It compares the tests using an illustrative example and a simulation study in which it examines the properties of tests for sparse contingency tables and compares them with frequently used tests. 1
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Hájek-Renyi inequality
Bělohlávek, Ivan ; Prášková, Zuzana (advisor) ; Čoupek, Petr (referee)
Title: A Hájek-Renyi inequality Author: Ivan Bělohlávek Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Zuzana Prášková, CSc., Department of Probability and Mathematical Statistics Abstract: In this thesis we study the Hájek-Rényi inequality for mixingales and their special cases. First, we prove the Hájek-Rényi inequality for martingales. Then, we investigate the relationship between the Kolomogorov and Hájek-Rényi inequalities. After that, we prove the law of large numbers using the Hájek-Rényi inequality. We then provide a detailed proof of maximal inequality for mixingales, which we then use to derive the Hájek-Rényi inequality for mixingales. We then apply the inequality to a multitude of special cases of mixingales. Keywords: Hájek-Rényi inequality, martingales, mixingales, linear process 1
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Serial correlations in time series
Kárný, Jakub ; Zichová, Jitka (advisor) ; Prášková, Zuzana (referee)
The subject of the thesis is the autocorrelation structure of time series. AR(1) process is studied as a special example. An estimator of the variance of the sample autocorre- lation is derived and its asymptotic properties are proved. We investigate the convergence of the variance estimates of sample autocorrelations in some simulated series. Further, the empirical significance level and power of selected autocorrelation tests are calculated. 1
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Cluster analysis methods and their applications in marketing
Dvořák, Marek ; Prášková, Zuzana (advisor)
In this work we study algorithms for cluster analysis and their application to the real data. In the beginning, the various types of data are presented. We define dissimilarity measures for each type of data and for clusters to be able to do the clustering and evaluate the separation quantitatively. In the Chapter 2, there are described partitioning algorithms and some criteria to determine the optimal number of clusters. A part of this chapter is devoted to the fuzzy cluster analysis which is a generalization of partitioning techniques. Hierarchical algorithms are characterized in Chapter 3 as well as criteria for choosing the appropriate method. In the very end of this chapter, there is a comparison of all the methods in terms of various types of the separation functionals. Archetypal analysis, which is another data mining instrument, is described in Chapter 4. All chapters include illustration examples of usage. The main application part is the last chapter of this diploma thesis and it's based on the lifestyle survey in the Czech republic.
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Kernel estimates of hazard function
Selingerová, Iveta ; Horová, Ivanka (advisor) ; Prášková, Zuzana (referee)
Kernel estimates of hazard function Abstract This doctoral dissertation is devoted to methods for analysis of censored data in survival analysis. The main attention is focused on the hazard function that reflects the instantaneous probability of the event occurrence within the next time instant. The thesis introduces two approaches for a kernel esti- mation of this function. In practice, the hazard function can be affected by other variables. The most frequently used model suggested by D. R. Cox is presented and moreover two types of kernel estimates to estimate a condi- tional hazard function are proposed. For kernel estimates, there is derived some statistical properties and proposed methods of bandwidths selection. The part of the thesis is extensive simulation study where theoretical results are verified and the proposed methods are compared. The last chapter of the thesis is devoted to an analysis of real data sets obtained from different fields.
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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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Multivariate financial time series models in portfolio optimization
Bureček, Tomáš ; Hendrych, Radek (advisor) ; Prášková, Zuzana (referee)
This master thesis deals with the modeling of multivariate volatility in finan- cial time series. The aim of this work is to describe in detail selected approaches to modeling multivariate financial volatility, including verification of models, and then apply them in an empirical study of asset portfolio optimization. The results are compared with the classical approach of portfolio optimization theory based on unconditional moment estimates. The evaluation was based on four known op- timization problems, namely minimization of variance, Markowitz's model, ma- ximization of the Sharpe ratio and minimization of CVaR. The output portfolios were compared by using four metrics that reflect the returns and risks of the port- folios. The results demonstrated that employing the multivariate volatility models one obtains higher expected returns with less expected risk when comparing with the classical approach. 1
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Nonparametric Nonlinearity Testing in Time Series
Dudlák, Oliver ; Zichová, Jitka (advisor) ; Prášková, Zuzana (referee)
The aim of this bachelor thesis is nonparametric nonlinearity time series testing by using Q-tests and BDS-test. We describe theoretically each of the tests and then use them on simulated and real historical data. For tested time series we firstly try to identify linear model ARMA(p,q). Then we apply the tests on the estimated white noise to test the assumption of independence or noncorrelation and verify the accuracy of identified model.
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Econometric methods of change detection
Dvoranová, Romana ; Prášková, Zuzana (advisor) ; Hušková, Marie (referee)
Detection of structural changes in time series is a topic with increasing pop- ularity among econometricians over the last decades. The main aim of this thesis was to review and compare the classical and modern econometric meth- ods of structural change detection and unit root testing. A recent method for testing a one-time break in at most linear trend function of a series without prior knowledge about the stationary or unit root nature of the error compo- nent proposed by Perron and Yabu (2009b) was studied. Subsequently, it was combined with the unit root test that allows for a break in trend proposed by Kim and Perron (2009) to examine the nature of the error component. All the methods for change detection and unit root testing were compared in a Monte Carlo simulation study that indicated significant improvement in power of the Perron-Yabu and Kim-Perron tests against most alternatives compared to the classical methods. However, all tests demonstrated poor performance in case of a quadratic trend function. Finally, the tests were employed in a practical ex- ample to examine the properties of the quarterly GDP time series of the Czech Republic. 1
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