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Robustification of statistical and econometrical regression methods
Jurczyk, Tomáš ; Víšek, Jan Ámos (advisor) ; Hlávka, Zdeněk (referee) ; Malý, Marek (referee)
Title: Robustification of statistical and econometrical regression methods Author: Mgr. Tomáš Jurczyk Department: Department of probability and mathematical statistics Supervisor: prof. RNDr. Jan Ámos Víšek CSc., IES FSV UK Praha Abstract: Multicollinearity and outlier presence are two problems of data which can occur during the regression analysis. In this thesis we are interested mainly in situations where combined outlier-multicollinearity problem is present. We will show first the behavior of classical methods developed for overcoming one of these problems. We will investigate the functionality of methods proposed as robust multicollinearity detectors as well. We will prove that proposed two-step procedures (in one step typically based on robust regression methods) are failing in outlier detection and therefore also multicollinearity detection, if the strong multicollinearity is present in the majority of the data. We will propose a new one-step method as a candidate for the robust detector of multicollinearity as well as the robust ridge regression estimate. We will derive its properties, behavior and propose the diagnostic tools derived from that method. Keywords: multicollinearity, outliers, robust detector of multicollinearity, ro- bust ridge regression 1
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Wavelet transform
Valter, Boris ; Hlávka, Zdeněk (advisor) ; Dvořák, Jiří (referee)
Wavelet transform is a term from signal analysis. It is mostly used in physics, but also in finance, where we can use it to find a trend in different financial data. In the first chapter we will describe two older methods of signal analysis: Fourier transform and short-time Fourier transform. In the second chapter we show, how wavelet transform works, derive frequently used algorithm for calculating discrete wavelet transform and at the end we show several practical examples. This thesis was considered to deepen the knowledge of time-frequency analysis of signals, for better understanding of the principle, how wavelet transform works, and for potential extending its use. Powered by TCPDF (www.tcpdf.org)
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GOF tests for gamma distribution
Klička, Petr ; Hlávka, Zdeněk (advisor) ; Kulich, Michal (referee)
The Bachelor thesis deals with the goodness of fit test for the Gamma distribution. Initially, we show several ways how to estimate the parameters of the Gamma distribution - firstly, the maximum likelihood estimator is presented, followed by estimator gained by the method of moments and fi- nally, we introduce the new estimator based on the sample covariance. The last estimator is used for constructing the goodness of fit test for the Gamma distribution. We define the test statistics V ∗ n to this test and its asymptotic normality is derived under the assumption of the null hypothesis. At the end of the thesis the simulations are realized to obtain the empirical size of the test for various values of parameter a and parameter b which equals one. 1
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Homoscedasticity Tests in a Linear Model
Vávra, Jan ; Komárek, Arnošt (advisor) ; Hlávka, Zdeněk (referee)
This thesis deals with testing the assumption of homoscedasticity in linear model, that is the assumption of constant variance of this model. There is plenty of such tests, but not all of them can be applied to specific model and not all of them reach satisfactory results under various circumstances. Thesis focuses on tests which can be derived on the basis of the asymptotic theory for maximum likelihood estimation, particularly the test theory with nuisance parameters. There are derived two basic tests, the first one in the situation of analysis of variance model and the second one in the situation when we allow the dependence of variance to concomitant quantities. In subsequent numerical studies there are examined characteristics of derived test statistics. Powered by TCPDF (www.tcpdf.org)
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Simultaneous confidence intervals dual to stepwise methods of multiple comparison
Moravec, Jan ; Komárek, Arnošt (advisor) ; Hlávka, Zdeněk (referee)
The central theme of this thesis is the construction of simultaneous confidence regions (SCR) corresponding to stepwise multiple comparison procedures (MCP). The first chapter is devoted to the theory of multiple comparisons, including the class of closed testing procedures which contains every MCP that strongly con- trols the familywise error rate. The second chapter is concerned with the gene- ral principle of construction of SCR corresponding to closed testing procedures. These general results are used in the third and the forth chapter for deriving the SCR corresponding to a subclass of closed testing procedures which are based on weighted Bonferroni tests. The SCR corresponding to the Holm, the Holm(W), the fixed-sequence and the fallback MCP are derived explicitly. The theoretical results are numerically illustrated on a bioequivalence study. In the fifth chapter we briefly discuss the SCR corresponding to the Hommel, the Hochberg and the step-down Dunnett MCP.
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Multivariate goodness-of-fit tests
Kuc, Petr ; Hlávka, Zdeněk (advisor) ; Antoch, Jaromír (referee)
In this thesis we introduce, implement and compare several multivariate goodness-of-fit tests. First of all, we will focus on universal mul- tivariate tests that do not place any assumptions on parametric families of null distributions. Thereafter, we will be concerned with testing of multi- variate normality and, by using Monte Carlo simulations, we will compare power of five different tests of bivariate normality against several alternati- ves. Then we describe multivariate skew-normal distribution and propose a new test of multivariate skew-normality based on empirical moment genera- ting functions. In the final analysis, we compare its power with other tests of multivariate skew-normality. 1
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Multivariate Normal Distribution
Ježo, Jakub ; Kulich, Michal (advisor) ; Hlávka, Zdeněk (referee)
Title: Multivariate Normal Distribution Author: Jakub Ježo Department: Department of Probability and Mathematical Statistics Supervisor: doc. Mgr. Michal Kulich, Ph.D., Department of Probability and Mathematical Statistics Abstract: This bachelor thesis deals with the multivariate normal distribution, distributions derived from it and relations between them. The definition and characterization of the n-dimensional multinormal distribution, derivation of its characteristic function and definition of the matrix normal distribution are shown at the beginning. Further this thesis looks at the properties of the multivariate normal distribution and examines the linear combinations of normal vectors, li- near combinations of normal matrices and theirs properties. After that the qua- dratic forms of matrices from the normal distribution are shown, which leads to the Wishart distribution, its properties and the analysis of multidimensional data based on it. At the end of the thesis, the combinations of random vec- tors and matrix from the normal distribution are examined, which results to the Hotelling distribution and its properties. The distribution and properties of the sample mean vector and sample covariance matrix of a random sample from n- dimensional multinormal distribution are presented in this thesis....
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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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Goodness of fit tests with nuisance parameters
Baňasová, Barbora ; Hušková, Marie (advisor) ; Hlávka, Zdeněk (referee)
This thesis deals with the goodness of fit tests in nonparametric model in the presence of unknown parameters of the probability distribution. The first part is devoted to understanding of the theoretical basis. We compare two methodologies for the construction of test statistics with application of empirical characteristic and empirical distribution functions. We use kernel estimates of regression functions and parametric bootstrap method to approximate the critical values of the tests. In the second part of the thesis, the work is complemented with the simulation study for different choices of weighting functions and parameters. Finally we illustrate the use and the comparison of goodness of fit tests on the example with the real data set. Powered by TCPDF (www.tcpdf.org)
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