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Stochastical inference in the model of extreme events
Dienstbier, Jan ; Picek, Jan (advisor) ; Jurečková, Jana (referee) ; Jarušková, Daniela (referee)
Title: Stochastical inference in the model of extreme events Author: Jan Dienstbier Department/Institute: Department of probability and mathematical statistics Supervisor of the doctoral thesis: Doc. RNDr. Jan Picek, CSc. Abstract: The thesis deals with extremal aspects of linear models. We provide a brief explanation of extreme value theory. The attention is then turned to linear models Yn×1 = Xn×pβp×1 + En×1 with the errors Ei ∼ F, i = 1, . . . , n fulfilling the do- main of attraction condition. We examine the properties of the regression quantiles of Koenker and Basset (1978) under this setting we develop theory dealing with extremal characteristics of linear models. Our methods are based on an approximation of the regression quantile process for α ∈ [0, 1] expanding older results of Gutenbrunner et al. (1993). Our result holds in [α∗ n, 1 − α∗ n] with a better rate of α∗ n → 0 than the other approximations described previously in the literature. Consecutively we provide an ap- proximation of the tails of regression quantile. The approximations of the tails enable to develop theory of the smooth functionals, which are used to establish a new class of estimates of extreme value index. We prove T(F−1 n (1 − knt/n)) is consistent and asymp- totically normal estimate of extreme for any T member of the class....
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Modelling dependence between hydrological and meteorological variables measured on several stations
Turčičová, Marie ; Jarušková, Daniela (advisor) ; Hlávka, Zdeněk (referee)
Title: Modelling dependence between hydrological and meteorological variables measured on several stations Author: Bc. Marie Turčičová Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Daniela Jarušková CSc., Czech Technical University in Prague, Faculty of Civil Engineering, Department of Mathematics Abstract: The aim of the thesis is to explore the dependence of daily discharge averages of the Opava river on high daily precipitation values in its basin. Three methods are presented that can be used for analyzing the dependence between high values of random variables. Their application on the studied data is also given. First it is the tail-dependence coefficient that measures the dependence between high values of two continuous random variables. The model for the high quantiles of the discharge at a given precipitation value was first determined non-parametrically by quantile regression and then parametrically through the peaks-over-threshold (POT) method. Keywords: extremal dependence, tail-dependence coefficient, quantile regression, peaks over threshold method
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Prediction of transformed time series
Polák, Tomáš ; Jarušková, Daniela (referee) ; Anděl, Jiří (advisor)
The aim of this thesis is to find prediction for non-linear transformation of time series. First, under certain assumptions regarding the original time series, the autocovariance function and spectral density of the transformed time series are studied. General theorems are applied to concrete ARMA processes. Then general formulas for predictions of the transformed time series, which do not require knowledge of the autocovariance function of the transformed series nor its spectral density are presented. These formulas are applied to three concrete transformations and explicit formulas for ARMA processes are derived. Three types of predictions (optimal, naive and linear) are compared in the terms of proportional increase of mean square prediction error. Explicit formulas for ARMA processes are verified by a simulation.
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