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Exact Inference In Robust Econometrics under Heteroscedasticity
Kalina, Jan ; Peštová, B.
The paper is devoted to the least weighted squares estimator, which is one of highly robust estimators for the linear regression model. Novel permutation tests of heteroscedasticity are proposed. Also the asymptotic behavior of the permutation test statistics of the Goldfeld-Quandt and Breusch-Pagan tests is investigated. A numerical experiment on real economic data is presented, which also shows how to perform a robust prediction model under heteroscedasticity. Theoretical results may be simply extended to the context of multivariate quantiles
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Robust optimization in classification and regression problems
Semela, Ondřej ; Kalina, Jan (advisor) ; Lachout, Petr (referee)
In this thesis, we present selected methods of regression and classification analysis in terms of robust optimization which aim to compensate for data imprecisions and measurement errors. In the first part, ordinary least squares method and its generalizations derived within the context of robust optimization - ridge regression and Lasso method are introduced. The connection between robust least squares and stated generalizations is also shown. Theoretical results are accompanied with simulation study investigating from a different perspective the robustness of stated methods. In the second part, we define a modern classification method - Support Vector Machines (SVM). Using the obtained knowledge, we formulate a robust SVM method, which can be applied in robust classification. The final part is devoted to the biometric identification of a style of typing and an individual based on keystroke dynamics using the formulated theory. Powered by TCPDF (www.tcpdf.org)
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Normality test of the gene expression data
Shokirov, Bobosharif ; Klebanov, Lev (advisor) ; Hušková, Marie (referee) ; Kalina, Jan (referee)
This thesis deals with a test of normality of gene expressions data. Based on characterization theorems of the normal distribution, the test of normality is replaced by a test of spherical uniformity. Due to strong correlations between the gene expression data, the normality test is conducted with $\delta$ sequences. A new characterization theorem of the normal distribution is proven. Based on that, the normality test is conducted using Kolmogorov's test statistic. The obtained characterization results for the normal distribution are extended to the complete type of distributions and based on that, a test is conducted to verify whether the distributions of the two data sets of the gene expressions belong to the same type. Powered by TCPDF (www.tcpdf.org)
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Models with categorical response
Faltýnková, Anežka ; Kalina, Jan (advisor) ; Kulich, Michal (referee)
This thesis concentrates on regression models with a categorical response. It focuses on the model of logistic regression with binary response and its generalization in which two models are distinguished: multinomial regression with nominal response and multinomial regression with ordinal response. For all three models separately, the Wald test and the likelihood ratio test are derived. These theoretical derivations are then used to calculate the test statistics for specific examples in statistical software R. The theory described in the thesis is illustrated by examples with small and large number of explanatory variables.
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The small sample size problem in gene expression tasks
Athanasiadis, Savvas ; Duintjer Tebbens, Erik Jurjen (advisor) ; Kalina, Jan (referee)
Charles University in Prague Faculty of Pharmacy in Hradec Králové Department of Biophysics and Physical Chemistry Candidate: Savvas Athanasiadis Supervisor: Jurjen Duintjer Tebbens Title of diploma thesis: The small sample size problem in gene expression tasks The thesis addresses classification of genes to tumor types based on their gene expression signatures. The number of variables (amino-acids) to be inves- tigated is typically very high (in the thousands) while it is expensive and time- consuming to analyze a high number of genes; usually at most tens of them are available. The combination of a small sample size with a large number of variables makes standard statistical classification methods inappropriate. The thesis focuses on a modification of a standard classification method, Fisher's linear discriminant analysis, for the case where the number of samples is smaller than the number of variables. It proposes an improved strategy to test this modified method with leave-one-out cross validation. Using so- called low rank updates of the involved covariance matrices, the computational costs of the cross validation process can be reduced by an order of magnitude. Memory demands are reduced as well.
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Multidimensional statistics and applications to study genes
Bubelíny, Peter ; Klebanov, Lev (advisor) ; Jurečková, Jana (referee) ; Kalina, Jan (referee)
Title: Multidimensional statistics and applications to study genes Author: Mgr. Peter Bubelíny Department: Department of probability and mathematical statistics Supervisor: prof. Lev Klebanov, DrSc., KPMS MFF UK Abstract: Microarray data of gene expressions consist of thousands of genes and just some tens of observations. Moreover, genes are highly correlated between themselves and contain systematic errors. Hence the magnitude of these data does not afford us to estimate their correlation structure. In many statistical problems with microarray data, we have to test some thousands of hypotheses simultaneously. Due to dependence between genes, p-values of these hypotheses are dependent as well. In this work, we compared conve- nient multiple testing procedures reasonable for dependent hypotheses. The common manner to make microarray data more uncorrelated and partially eliminate systematic errors is normalizing them. We proposed some new normalizations and studied how different normalizations influence hypothe- ses testing. Moreover, we compared tests for finding differentially expressed genes or gene sets and identified some interesting properties of some tests such as bias of two-sample Kolmogorov-Smirnov test and interesting behav- ior of Hotelling's test for dependent components of observations. In the end of...
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Political activity of Anselm of Canterbury
Kalina, Jan ; Suchánek, Drahomír (advisor) ; Drška, Václav (referee)
The thesis aims to describe Anselm's years as prior and abbot and his archiepiscopal career. Analyzing the years spent in the Norman monastery of Bec as a missionary and teacher in its school, the thesis notes the amount of knowledge and experiences which prepared Anselm for his archiepiscopal career. His intellectual qualities and theories are examined as well as some of his highly influential theological texts. Anselm also strove to spread the reforms of his teacher and mentor at Bec and his predecessor at Canterbury, Archbishop Lanfranc. Anselm's following archiepiscopal career spanned the reigns of two kings: William Rufus and Henry I. The study proves that the policies and attitudes of both rulers were quite different. Under the reign of William Rufus, Anselm tried to bring his ideal theoretical state of things into actuality, but the king resisted everything he attempted to do. With his death, Anselm's position changed rapidly and dramatically. Henry, on the other hand, excelled in the ability to work out a compromise. In the end, Anselm's archiepiscopal career concluded with cooperation between king and archbishop.
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Tests of statistical hypotheses in measurement error models
Navrátil, Radim ; Jurečková, Jana (advisor) ; Hušková, Marie (referee) ; Kalina, Jan (referee)
The behavior of rank procedures in measurement error models was studied - if tests and estimates stay valid and applicable when there are some measurement errors involved and if not how to modify these procedures to be able to do some statistical inference. A new rank test for the slope parameter in regression model based on minimum distance esti- mator and an aligned rank test for an intercept were proposed. The (asymptotic) bias of R-estimator in measurement error model was also investigated. Besides measurement errors the problem of heteroscedastic model errors was considered - regression rank score tests of heteroscedasticity with nuisance regression and tests of regression with nuisance heterosce- dasticity were proposed. Finally, in location model tests and estimates of shift parameter for various measurement errors were studied. All the results were derived theoretically and then demonstrated numerically with examples or simulations.
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Robust classification and discrimination
Rensová, Dita ; Kalina, Jan (advisor) ; Jonáš, Petr (referee)
This thesis is focused on classification methods and their robust alternatives. First, we recall the standard classification rules of linear and quadratic discrim- ination analysis. We also show some methods for estimating their probability of missclassification. Next we describe some existing robust multivariate estimators, their properties and computational algorithms. These estimators are consequently used to construct robust classification rules. Then, we describe the principal com- ponent analysis as a technique for dimension reduction. Again, we study methods for its robustification. Finally, we illustrate the usage of robust classification on both numerical simulations and real data. We also investigate the influence of the principal component analysis on classification results.
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Robust Regression Estimators: A Comparison of Prediction Performance
Kalina, Jan ; Peštová, Barbora
Regression represents an important methodology for solving numerous tasks of applied econometrics. This paper is devoted to robust estimators of parameters of a linear regression model, which are preferable whenever the data contain or are believed to contain outlying measurements (outliers). While various robust regression estimators are nowadays available in standard statistical packages, the question remains how to choose the most suitable regression method for a particular data set. This paper aims at comparing various regression methods on various data sets. First, the prediction performance of common robust regression estimators are compared on a set of 24 real data sets from public repositories. Further, the results are used as input for a metalearning study over 9 selected features of individual data sets. On the whole, the least trimmed squares turns out to be superior to the least squares or M-estimators in the majority of the data sets, while the process of metalearning does not succeed in a reliable prediction of the most suitable estimator for a given data set.
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