
Goodnessoffit tests for Pareto distribution
Oganesyan, Robert ; Hlávka, Zdeněk (advisor) ; Zichová, Jitka (referee)
Pareto model is widely used in insurance and reinsurance, as well as in finance (capital management, stock returns), and in modelling of natural disasters. As a motivation, we introduce a case study drawn from the pricing of reinsurance excess of loss contracts. In the first chapter of the thesis, we define four types of Pareto distributions and explore such properties as moments, parameter estimation and characterizations. In the follow ing chapter, we examine the theory of unbiased estimation in order to better understand the construction of goodnessoffit tests based on characterizations. Subsequently, we introduce an overview of some goodnessoffit tests. The final chapter focuses on simu lating the type I error rates and power of these tests, as well as conducting a comparative analysis. Finally, we return to the motivational example and complete the calculation of the price for the reinsurance coverage. 1


Asymmetric volatility modelling in finance
Ploužková, Karolína ; Zichová, Jitka (advisor) ; Vejmělka, Petr (referee)
This bachelor thesis deals with modelling volatility in finance. The aim of the thesis is to introduce the models that can be used to this purpose. We focus on the GARCH and EGARCH models. For both models, we present their definition and investigate stationarity, the existence of unconditional moments and the correlation structure. We also present the GED distribution that is used in volatility modelling. In the practi cal part of the thesis, we show process simulations for different choices of parameters, investigate the accuracy of the parameter estimates, and finally perform an application of the GARCH and EGARCH models to the logarithmic returns of the Apple stocks. 1


Multivariate Volatility Modeling
Jurák, František ; Zichová, Jitka (advisor) ; Cipra, Tomáš (referee)
This thesis deals with the formulation and estimation of the multivariate GARCH model. It mentions the various parameterizations of the multivariate GARCH model and discusses the relationships between them. The necessary and sufficient conditions for covariance stationarity of the multivariate GARCH model are presented, as is the maximum likelihood estimation of the parameters of the model. The thesis also includes estimation of the parameters of the bivariate GARCH(1,1) model for real time series using EViews. 1


Elementary statistical methods in psychological research
Benešová, Anna ; Omelka, Marek (advisor) ; Zichová, Jitka (referee)
This work deals with testing methods for a twosample problem and their subsequent application to data from the field of behavioural economics. The main question of the analysis is to determine the existence of loss aversion and the en dowment effect. This thesis presents tests suitable for analyzing the given data. Specifically, it focuses on testing the ratio and difference of means and the MannWhitney formulation for the Wilcoxon test. In the next sec tion, the underlying article and the research topic are introduced. In the end, the methods are applied to the provided data set. The existence of the endowment effect is confirmed in at least half of the cases.


Financial time series models and their software implementation
Kostárová, Aneta ; Zichová, Jitka (advisor) ; Hudecová, Šárka (referee)
This thesis deals with financial time series models and their implementation in soft ware products. The theoretical part of the thesis includes a description of the volatility models ARCH, GARCH, IGARCH, ARCHM, GARCHM, EGARCH and GJRGARCH and their basic properties. The practical part examines and describes the implementation of the volatility models in the software products Mathematica, EViews and R. Tutorials on the use of each function are included, along with descriptions of the software inputs and outputs in the form of illustrative examples on simulated data and their application to real data. 1


Complex random variables
Kovalčíková, Emma ; Zichová, Jitka (advisor) ; Večeř, Jan (referee)
This bachelor thesis deals with complex random variables and complex random vec tors. We introduce the complex normal distribution by deriving it from the multivariate normal distribution and we describe maximum likelihood estimators of mean and variance matrix of a vector with a complex normal distribution. We conclude the theoretical part by describing a test for the nullity of the mean of the complex normal distribution. The practical part of the thesis consists of a simulation study in which we generate realizations of random vectors with complex normal distribution. We investigate the behavior of the parameter estimates of the complex normal distribution and the properties of the test of nullity of the mean for different sample sizes and different numbers of samples. Finally, we compare the empirical distribution of the test statistics derived in the theoretical part of the paper with the corresponding theoretical distribution. 1

 

Nonlinearity detection in financial time series
Dudlák, Oliver ; Zichová, Jitka (advisor) ; Prášková, Zuzana (referee)
The aim of this master thesis is nonparametric and parametric nonlinearity testing in time series and its application on real financial data. From nonparametric tests, we describe a bispectral density test. Based on it, we can test symmetry and linearity of observed time series. Because of complex nature of the test we included the theory of complex random variable. From parametric tests, we introduce the RESET test and its modifications, Keenan test and F test. Considering the analogy between these tests and the test for submodel in linear model, we included basic theory of linear model and multivariate linear regression model. For both cases we performed a simulation study, where we observe frequencies of rejections of null hypothesis in both linear and nonlinear time series. When we get frequencies corresponding to the theoretical significance levels of the test, statistics we continue analyzing the real data.


Reverse mortgage
Korotkov, Daniil ; Mazurová, Lucie (advisor) ; Zichová, Jitka (referee)
At this moment, reverse mortgages are relatively new products on the Czech market and this thesis deals with their problematics. In this thesis, we describe the main risks related to reverse mortgages, namely, longevity risk and adverse evolution of property prices. Analyzing these risks we are modelling the underlying property prices, their future behavior along with studying the risk mo dels such as vector autoregression. In practical part, we focus on estimating the parameters of LeeCarter model to estimate the distribution of life expectancy. In conclusion we apply the results of the estimation to calculate various charac teristics of reverse mortgages using the simplied version of the model. 1


Mean estimation in normal distribution
Kaliská, Andrea ; Zichová, Jitka (advisor) ; Hlávka, Zdeněk (referee)
The bachelor's thesis deals with estimators of the nonconstant mean value in a specific probability model, which was taken from the article Estimating the Current Mean of a Normal Distribution which is Subjected to Changes in Time. Firstly, we describe this model in detail and we add proofs. We consider two estimators: the minimum variance linear unbiased estimator and the Bayes estimator in cases with at most one change, which is taken directly from the article. The thesis is concluded with a simulation study, focusing on the comparison of those estimators. 1
