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Mathematical Models for LGD
Rychnovský, Michal ; Zvára, Karel (referee) ; Charamza, Pavel (advisor)
The aim of the present work is to describe possible models for LGD estimation and to test them on the real data. Besides common linear and logistic regression models we aim to describe the methods using running and censored observations - based on the Cox model and the two-step regression. This work first briefly outlines the principle of the capital requirement according to the Basel II. Then, individual methods are described and finally applied to the real banking data.
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Extraction of information from multidimensional data and its visualization
Vansa, Tibor ; Zvára, Karel (referee) ; Ranocha, Pavel (advisor)
The presented thesis aims to describe and compare methods of extraction of information from high dimensional data, with both continuous and discrete variables. It is focused on principal component analysis, factor analysis, multidimensional scaling and correspondence analysis. The thesis includes generalization of multidimensional scaling and correspondence analysis to case with more random variables. Mutual relations of presented methods are studied. Presented methods are applied on real data and their implementation in statistical softwares R and SPSS is desribed.
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Paired tests with missing observations
Sionová, Kristýna ; Kulich, Michal (referee) ; Zvára, Karel (advisor)
The aim of this work is to find procedures for testing the equality of means of a bivariate normal distribution when observations are missing on one or both variables. One of the possible solutions is to discard the incomplete data and to use the paired t test. In this thesis we describe tests that enable using all available data. For the case of missing observations on one variable, the solutions of Banerjee, Welch and Mehta and Gurland are presented. For the case of missing observations on both variables, the solutions of Banerjee, Welch, Bhoj and the solution based on the maximal likelihood estimate are presented. We compare the solutions found to each other and to the paired t test on simulated data. The software R 2.5.1 is used for this purpose.
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Effect of measurement error on the shape of the regression function in nonlinear models
Drábková, Alena ; Zvára, Karel (referee) ; Kulich, Michal (advisor)
In this thesis we study the effect of regressors measured with an error on an estimated coefficients in a generalized linear model. We infer the true shape of the mean and of the variance function in the given model. We show that assumptions of a generalized linear model are not fulfilled universally if we use variables measured with an error. Despite this, the error-in-variable model can still be useful for testing dependence of original correct regressor. Further on in the thesis, the asymptotic values of coefficients are approximated, assuming g(E(Yi|Wi)) is a quadratic function. Examples for all results are provided through simulations.
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Multicollinearity
Dřizgová, Lucie ; Hlávka, Zdeněk (referee) ; Zvára, Karel (advisor)
In our work, we explored multicollinearity problem from a complex point of view - from diagnostic methods to the solving of the problems which are caused by the multicollinearity. We compared the Least Squares method with some alternative methods - Principal Component Regression, Partial Least Squares Regression and Ridge Regression on the theoretical basis. In the last section, we demonstrated all methods on practical example computed in the program R.
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Rounding procedure
Chudoba, Martin ; Zvára, Karel (referee) ; Lachout, Petr (advisor)
In the presented work, we are introduced to the problem of rounding. In the process of numerous human activities, we nd a need, due to various reasons, to divide nite number of objects according to given ratios which cannot be ful lled exactly. Few basic methods have been developed throughout the history, but they were usually designed ad hoc, without appropriate mathematical background. "Fair" method should enjoy several well de ned properties or not allow some undesirable phenomenona. To avoid such paradoxes, restriction to so-called divisor methods is necessary. Within this group, methods should be inspected under additional criteria - mainly the propensity to prefer larger or smaller subjects, bias. Webster's method (Sainte-Lag ue) stands out as the unique unbiased divisor method, also featuring other important qualities. Real data comparison of chosen methods' most important properties is the nal part of this thesis.
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