
Flexibility, Robustness and Discontinuities in Nonparametric Regression Approaches
Maciak, Matúš ; Hušková, Marie (advisor)
Thesis title: Flexibility, Robustness and Discontinuity in Nonparametric Regression Approaches Author: Mgr. Matúš Maciak, M.Sc. Department: Department of Probability and Mathematical Statistics, Charles University in Prague Supervisor: Prof. RNDr. Marie Hušková, DrSc. huskova@karlin.mff.cuni.cz Abstract: In this thesis we focus on local polynomial estimation approaches of an unknown regression function while taking into account also some robust issues like a presence of outlying observa tions or heavytailed distributions of random errors as well. We will discuss the most common method used for such settings, so called local polynomial Msmoothers and we will present the main statistical properties and asymptotic inference for this method. The Msmoothers method is especially suitable for such cases because of its natural robust flavour, which can nicely deal with outliers as well as heavytailed distributed random errors. Another important quality we will focus in this thesis on is a discontinuity issue where we allow for sudden changes (discontinuity points) in the unknown regression function or its derivatives respectively. We will propose a discontinuity model with different variability structures for both independent and dependent random errors while the discontinuity points will be treated in a...


Multivariate generalized autoregressive conditional heteroscedasticity models
Nováková, Martina ; Pešta, Michal (advisor) ; Maciak, Matúš (referee)
This master thesis deals with extension of the univariate GARCH model to multivari ate models. We present individual models and deal with methods of their estimation. Then we describe some statistical tests for diagnosting the models. We have programmed in the statistical software R one of them  the LingLi test. Afterwards we apply selected models to real data of stock market index S&P 500, stock market index Russell 2000 and stocks of crude oil. For the GOGARCH model, we compare all available estimation methods and show their differences. Then we compare the results of all models with each other and also with univariate models in terms of estimates of conditional variances, estimates of conditional correlations and also in terms of computational complexity. 1


The power of two sample tests
Rózsahegyi, Dominik ; Maciak, Matúš (advisor) ; Komárek, Arnošt (referee)
Twosample tests are one of commonly used statistical tools with which we make decisions if experimentally obtained data from different populations satisfy prespecified statement. Suitability of using them could be dedicated by the power of test, which states for probability of rejection of invalid statement. The reader gets to know with terms of hypothesis testing which are necessary for introduction of tests. The second chapter introduce tests which could be used when analysed data are complete. If some observations are not exactly known, we call them censored and it is more suitable to use tests listed in the third chapter. We estimate the power for tests in simulations and observe its behavior with different conditions. 1


Analysis of fluctuation of labourers
Zeman, Ondřej ; Maciak, Matúš (advisor) ; Komárek, Arnošt (referee)
The main goal of this thesis is to analyse the fluctuation of the employees in a well established Czech manufacturing company. Due to the GDPR regulations, the underlying company is kept anonymised in this thesis. The data were transformed into longitudinal data and the GEE methodology was used for the analysis of the fluctuation. In the first chapter, an introduction to the problem and a short description of the data is provided. The second chapter contains some theoretical description of the GEE methodology and the QIC information criterion. In the third chapter, multiple models for a binary and multinomial response are fitted to the data and their results are described in detail. This allows us to describe the influence of various factors to the fluctuation of the employees in the underlying company. 1


Doseresponse curves
Hezoučký, Martin ; Hlávka, Zdeněk (advisor) ; Maciak, Matúš (referee)
Title: Doseresponse curves Author: Martin Hezoučký Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Zdeněk Hlávka, Ph.D., Department of Probability and Mathematical Statistics Abstract: In this thesis, we deal with the process of research and development of new medical substances with a focus on statistical methods used to determine appropriate doses. For this purpose, we examine the doseresponse relationship. First, we describe a typical procedure for the development of a new drug. Second, we focus in detail on the MCPMod method. Third, we propose a new method based on the theory of gradual change models. This approach tests whether the administration of the drug has a significant effect. If so, the dose with desired effect is estimated using an appropriate model. Specifically, we provide an esti mate using linear, quadratic and Emax gradual change models. We also describe a construction of a confidence interval for the point of change and also for the dose with the desired effect. The advantage of the proposed method over the MCPMod is the determination of the confidence intervals. Finally, we apply the above mentioned methods to data from the U.S. Tox21 research program and compare the results based on several tested substances and clearly demonstrate the...


Twosample Wilcoxon test in the presence of ties
Kulla, Filip ; Omelka, Marek (advisor) ; Maciak, Matúš (referee)
This work is devoted to the Wilcoxon ranksum test in the presence of ties. In this work, asymptotic distribution of the Wilcoxon test statistic in the presence of ties is derived using wellknown results about Ustatistics. With the aid of this result, a corrected test statistic for data containing ties is proposed. Furthermore, the thesis examines the relation between the derived correction and conditional variance of the test statistic (and conditional expectation). Finally, by means of simulation, the effect of the derived correction on the actual significance level is examined as the number of tied observations increases. 1


Semiparametric additive risk model
Zavřelová, Adéla ; Kulich, Michal (advisor) ; Maciak, Matúš (referee)
Cox proportional hazard model is often used to estimate the effect of covariates on hazard for censored event times. In this thesis we study the semiparametric models of additive risk for censored data. In this model the hazard is given as a sum of unknown baseline hazard function and a product of covariates and coefficients. Further the general additivemultiplicative model is assumed. In this model the effect of a covariate can be either multiplicative, additive or both at the same time. We focuse on determining the effect of a covariate in the general model. This model can be used to test for the multiplicative or addtive effect of a covariate on the hazard.


L1 Regression
Čelikovská, Klára ; Maciak, Matúš (advisor) ; Hlubinka, Daniel (referee)
This thesis is focused on the L1 regression, a possible alternative to the ordinary least squares regression. L1 regression replaces the least squares estimation with the least absolute deviations estimation, thus generalizing the sample median in the linear regres sion model. Unlike the ordinary least squares regression, L1 regression enables loosening of certain assumptions and leads to more robust estimates. Fundamental theoretical re sults, including the asymptotic distribution of regression coefficient estimates, hypothesis testing, confidence intervals and confidence regions, are derived. This method is then compared to the ordinary least squares regression in a simulation study, with a focus on heavytailed distributions and the possible presence of outlying observations. 1


The power of two sample tests
Rózsahegyi, Dominik ; Maciak, Matúš (advisor) ; Nagy, Stanislav (referee)
Twosample tests are commonly used in practice, for example in scientific sphere or financial sectors. The power of test is also an important feature and express probability that the test will reject invalid null hypothesis. In this work we introduce four basic tests in which we compare some parameters of two popu lations. The reader gets to know with basic terms of hypothesis testing which are necessary for introduction of tests. For each test we use simulation for estima tion of the power and observe its behavior with different distributions of samples, ranges or selected null and alternative hypothesis. Based on obtained results we compare chosen tests and discuss suitability for using them in different cases. 1


Generalized Wilcoxon Test for Censored Data
Vařejková, Michaela ; Maciak, Matúš (advisor) ; Komárek, Arnošt (referee)
This paper deals with the generalized Wilcoxon test and its use for censored data. The introduction describes standard onesample and twosamples Wilco xon tests and their basic properties, censored data and methods of censoring. The main part of the paper is devoted to the introduction of the generalized Wilcoxon test and to its properties. First, a test for singlycensored data is de scribed; the description of a test for doubly censored data follows. The paper concludes with a simulations part in which statistical properties of the test are demonstrated. The first example compares the generalized test with the stan dard twosamples Wilcoxon test. The second example shows how the censoring rate affects the power and significance level of the generalized test. 1
