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Tests for Multiple Changes in Linear Regression Models
Marušiaková, Miriam ; Hušková, Marie (advisor) ; Prášková, Zuzana (referee) ; Picek, Jan (referee)
We consider tests for multiple structural changes in linear regression models. The tests are based on F-type test statistics for the null hypothesis of no change against k changes or against an unknown number of changes with a given upper bound. We extend the existing results to linear regression models with deterministically trending regressors. Moreover, we introduce a generalized M-type test statistic which is based on functionals of weighted M-residuals. In change-point analysis approximations to critical values are usually obtained through the limit behavior of the respective test statistic under the null hypothesis. However, these approximations are often not satisfactory. Either the convergence of the test statistic to its limit distribution is rather slow or the limit distribution itself is very complex. An alternative approach is to apply resampling methods. We explore this possibility for F-type and M-type test statistics in the presence of multiple change points. We prove that the bootstrap method provides asymptotically correct critical values for the studied tests. We conduct several simulation experiments to show that the bootstrap based approximations are reasonable also in nite sample situations. Moreover, these approximations are often better than the asymptotic critical values. Finally, we...
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Multivariate ARCH and GARCH models
Šafránková, Jana ; Hurt, Jan (referee) ; Prášková, Zuzana (advisor)
We study multivariate ARCH and GARCH models and their subsequent application to simulated and real data. In discussed models the conditional variance matrix is considered to be a function of lagged data process which is the subject of study. In case of GARCH models the conditional variance matrix is dependent on own lagged values, too. First of all, we deal with univariate ARCH and GARCH models to get some theoretical basis. The subsequent study extends this basis to multivariate models. A survey of multivariate GARCH models is presented in the next part of this thesis. Further study is devoted to maximum likelihood estimators of these models and we deal with alternatives to multivariate normal distribution which is a standard assumption of this method. We occupy ourselves with tests of these models, too. We mention both preestimation tests and postestimation tests to verify the adequacy of models. In conclusion we give practical examples which show di±culties of applications of these models for real data.
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Decomposition methods for time series with irregular observations
Hanzák, Tomáš ; Prášková, Zuzana (referee) ; Cipra, Tomáš (advisor)
This work deals with extensions of classical exponential smoothing type methods for univariate time series with irregular observations. Extensions of simple exponential smoothing, Holt method, Holt-Winters method and double exponential smoothing which have been developed in past are presented. An alternative method to Wright's modification of simple exponential smoothing for irregular data, based on the corresponding ARIMA process, is suggested. Exponential smoothing of order m for irregular data as a generalization of simple and double exponential smoothing is derived. A similar method using a DLS (discounted least squares) estimation of polynomial trend of order m is derived as well. In all cases the recursive character of these methods is preserved making them easy to implement and high computationally effective. A program in which most of the methods presented here are available is a part of the work. Some numerical examples of their application are also included.
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Applications of Markov processes to modeling HIV disease progression
Žohová, Ivana ; Prášková, Zuzana (referee) ; Kulich, Michal (advisor)
Nazev prace: Aplikace Mnrkovovych procesu pfi modclovaiii prubehu cho- n.hy HIV Autor: I\aua Zohova Katcdra: Katcdra pravdepodobnosli a malematicke statistiky Vedouci bakalafske prace: Mgr. Michal Ivulich,Ph.D. o-mail vedonciho: kulich fikarlm.mff'.cnni.c/ Abstrakt: V pfedio/ene praci se zabyvame slochastickym modelovam'm priibellii choroby HIV. pfedevshu pak inkubacni doln' AIDS. Hlavni po/or- noyt jc vuiiuvana. inorldlovdni |)oniuc( hoino^eiiiiicli Markovovych proce.sit sc s])ojit.yni c;isein a kouernou iniioziiion stavu. V praci ncjprvc Ktuclujoiric nOklora .spojili'i ro'/drloni, a to pft'devsiin z hk'diska jcjich vliodnosl.i k ino- dolovani inkulmrnf doby AIDS. Dalsi' cast jc vonovana konstnikci Ma.rko- vuvych niodelu. Pnilioh fihoroliv HIV jc iicjprvr niixlrlovaii poinorf dvou Iconkrntnicli inodrlu a jjotc pomoci zdliccnoDrhc) modclu. V niodelcch JHOTI odvox.cna. rozdrlrni inkubariii duljy AIDS. Klicova slova: chornba HIV. inkiiliariii doba AIDS. lhmiu»t'iim' Markovuv l>ro('{'s. Markovuv Title: Ap])lications of Markov processes to modeling, HIV di.scas siun Author: Ivana Zohuva Department: Drpartnicnt of Probability and MaHicniatical Statistics Supervisor: Mgr. Michal Kulich, Ph.D. Supervisor's e-mail address: kuhch^karhn.nifF.riirii.c? Abstract: In tlu1 present work \vc study stochastic modeling of...
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Covariance function of space-time series
Tetíková, Vendula ; Prášková, Zuzana (referee) ; Cipra, Tomáš (advisor)
Space-time series and their covariance function are very important in areas such as meteorology, hydrology and evolutionary biology. This diploma thesis concentrates on space-time covariance functions and on the way how this covariance functions describe space-time interaction. It focuses mainly on stationary nonseparable covariance functions but also on separable ones. The aim is to introduce main methods of creation covariance functions. In practice the estimating of covariance function is different, therefore there are, at the conclusion, examples of solutions and their comparison.
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Long memory processes
Prelcová, Zuzana ; Cipra, Tomáš (referee) ; Prášková, Zuzana (advisor)
Cílem této práce je zasvětit čtenáře do problematiky procesů a s dlouhou pamětí. V úvodu je čtenář nejprve seznámen s tím, proč již procesy s krátkou pamětí, např. ARMA procesy, nepostačovali statistikům na modelování časových řad. V první kapitole je ukázán vývoj, který vedl k definici procesů a dlouhou pamětí. Rovněž jsou zde zmíněné stochastické procesy, které se používají na modelování dlouhodobé závislosti. Následně se čtenář obeznámí se základními metodami pro odhad parametru dlouhé paměti, s testy dobré zhody a spůsobmi predikce budoucích pozorování. Hlavním cílem této práce je porovnat metody na odhad parametru dlouhé paměti, které vznikly za posledných 15 let, na reálnych datech a ukázat jejich výhody či patřičné nedostatky.
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