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Modely celočíselných časových řad s náhodnými koeficienty
Burdejová, Petra ; Prášková, Zuzana (advisor) ; Cipra, Tomáš (referee)
Title: Models of integer-valued time series with random coefficients Author: Petra Burdejová Department: Department of Probability and Mathematical Statistics Supervisor: Doc. RNDr. Zuzana Prášková, CSc. Abstract: In the presented thesis, a generalized integer-valued autoregres- sive process of the order p (GINAR(p)) is considered first. The main aim is taken to introduction of random coefficient integer-valued autoregressive process (RCINAR(p)). We use a thinning operator in order to define the processes. The main characteristics of GINAR(p) and RCINAR(p) are obtained. Condi- tions for stationarity and ergodicity are stated. Three methods of estimation (Yule-Walker, Conditional least squares, Generalized method of moments) are given and compared in simulation with respect to the mean squared error (MSE). At the end, RCINAR(3) model is applied to a real dataset representing a number of earthquakes per year. Keywords: thinning operator, random coefficients, integer-valued time se- ries, GINAR, RCINAR
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Boostrapping Markov Chains
Marko, Dominik ; Prášková, Zuzana (advisor) ; Hušková, Marie (referee)
In this thesis we deal with estimating the transition matrix probabilities of discrete time Markov chains with finite state space. We will use two methods, specifically the maximum likelihood method and the bootstrap method, for obtaining estimators of these matrix probabilities and then we will develop the asymptotic distribution of these estimators. We will describe the basic characteristics of the bootstrap method and show the application of two bootstrap methods used for estimating transition probabilites, specifically conditional bootstrap and standard bootstrap. The results of the application of every method used for obtaining transition probabilities and computing confidence intervals will be presented in a numerical study and compared with the results based on asymptotic normality. Powered by TCPDF (www.tcpdf.org)
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Financial time series modelling with trend
Studnička, Václav ; Zichová, Jitka (advisor) ; Prášková, Zuzana (referee)
Various models can be used for the analysis of financial time series. This thesis focuses mainly on two models; non-linear trend model and linear trend model. First chapter is theoretial, there is an introduction to the theory of time series and to the autoregressive process. Second chapter is also theoretical and it focuses on a description of both non-linear and linear trend model including derivations of im- portant properties of these models; moreover, it contains theory for the modelling of financial time series and predictions. Last chapter contains simulations of two mentioned models and estimations of their parameters, Wolfram Mathematica is used for all simulations. 1
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Usage of Markov chains in banking
Klímová, Hana ; Marada, Tomáš (advisor) ; Prášková, Zuzana (referee)
The aim of the thesis is to get acquainted with the theory of Markov chains and to show how it is used in banking for estimation of credit rating transitions. In the first part, an introduction to the theory of discrete-time and continuous-time Markov chain with discrete state space is provided. In the next part three estimating methods that are used to calculate credit rating transitions - namely cohort method, durability method and Aalen-Johansen estimator are described theoreticaly. In the last part these methods are applied to calculate the matrices of transition probabilities on the basis of real rating migrations. Next an empirical transition matrix is used to simulate set of rating progressions, which are then used for estimating the original matrix by all the above mentioned methods. Finally the distance between the original and estimated matrices is evaluated to show the differences between the methods.
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Testing Structural Changes Using Ratio Type Statistics
Peštová, Barbora ; Hušková, Marie (advisor) ; Prášková, Zuzana (referee) ; Jarušková, Daniela (referee)
Testing Structural Changes Using Ratio Type Statistics Barbora Peštová Charles University in Prague, Faculty of Mathematics and Physics, Department of Probability and Mathematical Statistics, Czech Republic Abstract of the doctoral thesis We deal with sequences of observations that are naturally ordered in time and assume various underlying stochastic models. These models are parametric and some of the parameters are possibly subject to change at some unknown time point. The main goal of this thesis is to test whether such an unknown change has occurred or not. The core of the change point methods presented here is in ratio type statistics based on maxima of cumulative sums. Firstly, an overview of thesis' starting points is given. Then we focus on methods for detecting a gradual change in mean. Consequently, procedures for detection of an abrupt change in mean are generalized by considering a score function. We explore the possibility of applying the bootstrap methods for obtaining critical values, while disturbances of the change point model are considered as weakly dependent. Procedures for detection of changes in parameters of linear regression models are shown as well and a permutation version of the test is derived. Then, a related problem of testing a change in autoregression parameter is studied....
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Statistical Problems in Markov Chains
Adamová, Markéta ; Prášková, Zuzana (advisor) ; Branda, Martin (referee)
In this thesis we study basic statistical methods in Markov chains. In the case of discrete time, this thesis is focused on estimation of transition probability matrix and some basic tests (test for a specified transition probability matrix, test for homogeneity, test for independence, test for the order of Markov chain). In the case of continuous time we will concentrate on Poisson process and birth and death process. Estimation of parameters of these processes and tests for processes with specified parameters are mentioned. Developed estimates and test statistics are applied to real data in the final chapter.
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Tests of significance of ARMA models parameters based on Bayesian approach
Onderko, Martin ; Krtek, Jiří (advisor) ; Prášková, Zuzana (referee)
This thesis is focused on Bayesian analysis and its use in probability and statistics. It also marginally discusses random processes, furtherly describes ARMA model and defines the issue of estimation of the parameters of Bayesian approach. Acquired knowledge and derived characteristics subsequently applies in testing of significance of parameters. Thus it undoubtably affects the area of hypothesis testing and serves mainly as a tool to determine the ARMA model more accurately. This work should be regulary applied when detecting the necessity of testing of statistical significance of parameters of ARMA model.
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Principal components analysis and its applications
Dubová, Mária ; Hendrych, Radek (advisor) ; Prášková, Zuzana (referee)
In the present thesis, we deal with the principal components analy- sis. In the first of this text, we study different aspects of principals components, for instance, their derivation for a multidimensional random vector from general distribution or their calculation based on a covariance or correlation matrix. It is also important to choose the proper number of principal components for reducing the dimensionality of data in order to preserve most of information. Theoretical knowledge are illustrated with several examples. In the second part of the thesis, we focus on the value at risk. This term is defined in the text also with seve- ral usual formulas to calculate it. Then, we deal with a practical application of this concept and the principal component analysis. Concretely, we analyse the portfolio of some different interest rates to obtain the value at risk in some cases. 1
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Compound Poisson distribution
Valentovičová, Katarína ; Hudecová, Šárka (advisor) ; Prášková, Zuzana (referee)
Claims reserving and claims process estimation are classical problems in general insurance. Some of the statistical methods in this field are based on a compound distribution. This distribution arises as a sum of a random number of independent and identically distributed variables. This thesis deals, in particular, with the compound Poisson distribution, its properties and possible applications in general insurance. Basic theoretical properties of the distribution are derived, and parameters estimation methods are discussed. The theoretical methods are illustrated on a real data set from car insurance.
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