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Nonlienar volatility modeling in financial time series
Sychova, Maryna ; Zichová, Jitka (advisor) ; Hlávka, Zdeněk (referee)
In this work we want to examine selected models with nonlinear volatility and their properties. At the beginning we define models with non-constant variance, especially ARCH, GARCH and EGARCH models. Then we study the probability distributions that are mainly used in the EGARCH model. Then we focus on the EGARCH model, describe the conditions for stationarity and invertibility of the model, define diagnostic tests and QMLE estimates of parameters. In the last chapter we perform simulation studies of the selected models and their application to real data. 1
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Distribution of interpoint distances
Horská, Šárka ; Hlávka, Zdeněk (advisor) ; Komárek, Arnošt (referee)
This thesis investigates basic properties of the interpoint distances be- tween random vectors drawn from multinomial distribution. We also describe a possible application to testing sparse observations, i.e., a setup with small number of observations and large number of categories, where the classical χ2 -test cannot be recommended. As an alternative, utilizing the multinomial interpoint distances, we will present the test statistic proposed by Biswas and Ghosh (2014). 1
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Dose-response curves
Hezoučký, Martin ; Hlávka, Zdeněk (advisor) ; Maciak, Matúš (referee)
Title: Dose-response 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 dose-response relationship. First, we describe a typical procedure for the development of a new drug. Second, we focus in detail on the MCP-Mod 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 MCP-Mod 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...
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Gradual change model
Míchal, Petr ; Hlávka, Zdeněk (advisor) ; Pešta, Michal (referee)
The thesis aims at change-point estimation in gradual change models. Methods avail- able in literature are reviewed and modified for point-of-stabilisation (PoSt) context, present e.g. in drug continuous manufacturing. We describe in detail the estimation in the linear PoSt model and we extend the methods to quadratic and Emax model. We describe construction of confidence intervals for the change-point, discuss their interpre- tation and show how they can be used in practice. We also address the situation when the assumption of homoscedasticity is not fulfilled. Next, we run simulations to calculate the coverage of confidence intervals for the change-point in discussed models using asymp- totic results and bootstrap with different parameter combinations. We also inspect the simulated distribution of derived estimators with finite sample. In the last chapter, we discuss the situation when the model for the data is incorrectly specified and we calculate the coverage of confidence intervals using simulations. 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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Comparison of Models for Probabilities in Football Betting
Kožnar, František ; Večeř, Jan (advisor) ; Hlávka, Zdeněk (referee)
The aim of the thesis is to compare different statistical models for football betting odds and determine the best performing once based on the historical performance of sport teams. There are at least two possible approaches for computing the odds, namely Poisson regression and methods based on statistical machine learning. The idea is that the historical performance of teams is a good predictor of the future performance. Thus we can take the past performances, say all matches in the full season of the Bundesliga (306 matches), and use these data for predicting the odds for the following season. The resulting odds should be compared with the actual results using the scoring rules, which will identify the best performing model. 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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Comparison of Models for Probabilities in Football Betting
Kožnar, František ; Večeř, Jan (advisor) ; Hlávka, Zdeněk (referee)
The aim of the thesis is to compare different statistical models for football betting odds and determine the best performing once based on the historical performance of sport teams. There are at least three possible approaches for computing the odds, namely logistic regression, Poisson regression and methods based on statistical machine learning. The idea is that the historical performance of teams is a good predictor of the future performance. Thus we can take the past performances, say all matches in the full season of the English Premier League (380 matches), and use these data for predicting the odds for the following season. The resulting odds should be compared with the actual results using the scoring rules, which will identify the best performing model.
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Statistical machine learning with applications in music
Janásková, Eliška ; Večeř, Jan (advisor) ; Hlávka, Zdeněk (referee)
The aim of this thesis is to train a computer on Beatles' songs using the re- search project Magenta from the Google Brain Team to produce its own music, to derive backpropagation formulas for recurrent neural networks with LSTM cells used in the Magenta music composing model, to overview machine learning techniques and discuss its similarities with methods of mathematical statistics. In order to explore the qualities of the artificially composed music more thor- oughly, we restrict ourselves to monophonic melodies only. We train three deep learning models with three different configurations (Basic, Lookback, and At- tention) and compare generated results. Even though the artificially composed music is not as interesting as the original Beatles, it is quite likeable. According to our analysis based on musically informed metrics, artificial melodies differ from the original ones especially in lengths of notes and in pitch differences be- tween consecutive notes. The artificially composed melodies tend to use shorter notes and higher pitch differences. 1
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