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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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Statistical inference based on saddlepoint approximations
Sabolová, Radka ; Jurečková, Jana (advisor) ; Hlávka, Zdeněk (referee) ; Picek, Jan (referee)
Title: Statistical inference based on saddlepoint approximations Author: Radka Sabolová Abstract: The saddlepoint techniques for M-estimators have proved to be very accurate and robust even for small sample sizes. Based on these results, saddle- point approximations of density of regression quantile and saddlepoint tests on the value of regression quantile were derived, both in parametric and nonpara- metric setup. Among these, a test on the value of regression quantile based on the asymptotic distribution of averaged regression quantiles was also proposed and all these tests were compared in a numerical study to the classical tests. Finally, special case of Kullback-Leibler divergence in exponential family was studied and saddlepoint approximations of the density of maximum likelihood estimator and sufficient statistic were also derived using this divergence. 1
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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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Frequency analysis of precipitation amounts
Rulfová, Zuzana ; Kyselý, Jan (advisor) ; Holtanová, Eva (referee) ; Picek, Jan (referee)
Title: Frequency analysis of precipitation amounts Author: Mgr. Zuzana Rulfová Department: Department of Atmospheric Physics Supervisor: RNDr. Jan Kyselý, Ph.D., Institute of Atmospheric Physics CAS Abstract: This thesis deals with analysing characteristics of mean and extreme precipitation in observations and regional climate models (RCMs) with respect to their convective and stratiform origin. An algorithm for subdivision of precipitation amounts into predominantly convective and stratiform using station weather data is proposed and evaluated. The time series of convective and stratiform precipitation from the Czech Republic over 1982-2010 are used for analysing basic climatological characteristics of precipitation, including extremes, and evaluating RCMs from the ENSEMBLES project. Projected changes of convective and stratiform precipitation in Central Europe (the Czech Republic) are analysed using data from RCM simulations from the EURO-CORDEX project. The last part of the thesis introduces a new statistical model for analysing precipitation extremes. This model takes advantage from knowledge of origin of precipitation extremes. In future climate we could expect more convective and stratiform precipitation amounts in all seasons except summer, when climate models project decline in amounts of stratiform...
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Parameter Estimation under Two-phase Stratified and Cluster Sampling
Šedová, Michaela ; Kulich, Michal (advisor) ; Picek, Jan (referee) ; Omelka, Marek (referee)
Title: Parameter Estimation under Two-phase Stratified and Cluster Sampling Author: Mgr. Michaela Šedová Department: Department of Probability and Mathematical Statistics Supervisor: Doc. Mgr. Michal Kulich, Ph.D. Abstract: In this thesis we present methods of parameter estimation under two-phase stratified and cluster sampling. In contrast to classical sampling theory, we do not deal with finite population parameters, but focus on model parameter inference, where the ob- servations in a population are considered to be realisations of a random variable. However, we consider the sampling schemes used, and thus we incorporate much of survey sampling theory. Therefore, the presented methods of the parameter estimation can be understood as a combination of the two approaches. For both sampling schemes, we deal with the concept where the population is considered to be the first-phase sample, from which a sub- sample is drawn in the second phase. The target variable is then observed only for the subsampled subjects. We present the mean value estimation, including the statistical prop- erties of the estimator, and show how this estimation can be improved if some auxiliary information, correlated with the target variable, is observed for the whole population. We extend the method to the regression problem....
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Visualization of DTMs made from airborne laser scanning data
Picek, Jan ; Lysák, Jakub (advisor) ; Dušánek, Petr (referee)
Visualization of digital terrain models made from airborne laser scanning data Abstract The aim of this study is to evaluate opportunities of 2D visualization of digital terrain models made from airborne laser scanning data. Each method is described in detail with other parameters and approach. The focus is put on methods which are useful for the topographic mapping (f. e. orienteering maps) and for cartographic visualization. All this methods are tested for different data and type of relief (plain, mountainous, rock city, area with negative shapes). The aim of study is to find the optimal method for each type of relief which expresses most of typical features. There are shown methods on the maps for orienteering sports for detection of different object from airborne laser scanning data. Keywords: airborne laser scanning, digital terrain model, data visualization, hillshading
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Production of cartographically correct countour lines from airborne laserscanning data in flat terrain
Picek, Jan ; Lysák, Jakub (advisor) ; Bayer, Tomáš (referee)
Production of cartographically correct countour lines from airborne laserscanning data in flat terrain Abstract The thesis is dedicated to the creation of contour lines from the data of airborne laserscanning in flat areas. For creation of contour lines in these parts is necessary specific generalization, that smooths unwanted artifacts, but does not delete the shapes, which are characteristic for the relief. The first part of the thesis is to serve as a short insight into the areas on which it is built, this thesis: the principle of airborne laserscanning and basic information on the contour line model. On the basis of technical literature are defined the requirements for the correct representation of the contour and cartographicly presented the different types of the DEM and generalization of contours. The main aim of this work is to design an algorithm that will combine several methods of generalization of contour while maintaining altitude accuracy as well as other requirements on their cartographicly the right terms, and to test this approach in several territories. In conclusion, the results of the methods of work and its implementation critically evaluated and outlined other possible developments of this issue. Keywords hypsography, airborne laserscanning, digital cartography, contour lines,...
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Analysis of wind gusts over the area of the Czech Republic
Pop, Lukáš ; Sokol, Zbyněk (advisor) ; Picek, Jan (referee) ; Pelikán, Emil (referee)
The Ph.D. thesis deals with extreme wind gust analysis over the area of the Czech Republic. The first part of the thesis deals with processing of wind measurements, in particular maximum wind gusts measurements. Analysis of high-frequency wind measurement using 3-D sonic anemometer on the Kopisty station is included. Homogenization of the highest daily wind gusts was performed. Descriptive statistical analysis of measured wind gust values was performed. The following part of the thesis describes statistical theory of extreme values and discusses its applicability to wind gust data. Some theoretical findings were obtained. Numerous numerical experiments were performed focused on evaluation of proposed method. In the last part of the thesis station measurements were processed using the proposed methods and a model of dependence between extreme and mean wind climate was derived. The model was applied to the map of mean wind climate calculated earlier on the Institute of Atmospheric Physics and thus a map of extreme wind climate was obtained. The accuracy of this map was estimated. The map was compared with other maps of extreme wind calculated by other authors earlier.
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