
Total Least Squares and Their Asymptotic Properties
Chuchel, Karel ; Pešta, Michal (advisor) ; Antoch, Jaromír (referee)
Tato práce se zabývá metodou úplně nejmenších čtverc·, která slouží pro odhad parametr· v lineárních modelech. V práci je uveden základní popis metody a její asymptotické vlastnosti. Je vysvětleno, jakým zp·sobem lze v konceptu metody využít neparametrický bootstrap pro hledání odhadu. Vlastnosti bootstrap od had· jsou pak simulovány na pseudo náhodně vygenerovaných datech. Simulace jsou prováděny pro dvourozměrný parametr v r·zných nastaveních základního modelu. Jednotlivé bootstrap odhady jsou v rovině řazeny pomocí Mahalanobis a Tukey statistical depth function. Simulace potvrzují, že bootstrap odhad dává dostatečně dobré výsledky, aby se dal využít pro reálné situace.


Profit Maximization of Car Manufacturers Facing EU CO2 Emission Penalties From 2021
Leamer, Anthony David ; Večeř, Jan (advisor) ; Antoch, Jaromír (referee)
Title: Profit maximization of car manufacturers facing EU CO2 emission penalties from 2021 Author: Anthony David Leamer Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Jan Večeř, Ph.D., Department of Probability and Mathematical Statistics Abstract: This paper sheds light on the newly coming emissions penalization sys tem imposed on passenger vehicles registered in the EU. We analyze the penalty based on how it influences profit of the car manufacturers. After optimizing the profit margins car manufacturers impose on different vehicles we discuss what this means for the consumer and the manufacturer. We seek to answer the ques tion 'Who is going to pay the penalty?'. In the last chapter we analyze real data to see if the penalty will motivate manufacturers to produce more ecofriendly passenger cars. The data shows that the manufacturers will lose profit until the fleets' average emissions fall within the limits. The maximization apparatus developed in this paper is indeed standard  in the sense that there are no new theories developed  although the problem is new to the extent that it requires new creative use of specific parts of optimization theory. Moreover the decision of the EU to implement drastic measures to bring down, 'on road CO2 emissions', leads...


Big data  extraction of key information combining methods of mathematical statistics and machine learning
Masák, Tomáš ; Antoch, Jaromír (advisor)
This thesis is concerned with data analysis, especially with principal component analysis and its sparse modi cation (SPCA), which is NPhardto solve. SPCA problem can be recast into the regression framework in which spar sity is usually induced with ℓ1penalty. In the thesis, we propose to use iteratively reweighted ℓ2penalty instead of the aforementioned ℓ1approach. We compare the resulting algorithm with several wellknown approaches to SPCA using both simulation study and interesting practical example in which we analyze voting re cords of the Parliament of the Czech Republic. We show experimentally that the proposed algorithm outperforms the other considered algorithms. We also prove convergence of both the proposed algorithm and the original regressionbased approach to PCA. vi


Geometric approach to the estimation of scatter
Bodík, Juraj ; Nagy, Stanislav (advisor) ; Antoch, Jaromír (referee)
In this thesis we describe improved methods of estimating mean and scatter from multivariate data. As we know, the sample mean and the sample variance matrix are nonrobust estimators, which means that even a small amount of measurement errors can seriously affect the resulting estimate. We can deal with that problem using MCD estimator (minimum covariance determinant), that finds a sample variance matrix only from a selection of data, specifically those with the smallest determinant of this matrix. This estimator can be also very helpful in outlier detection, which is used in many applications. Moreover, we will introduce the MVE estimator (minimum volume ellipsoid). We will discuss some of the properties and compare these two estimators.


The method of reweighting (calibration) in survey sampling
Michálková, Anna ; Omelka, Marek (advisor) ; Antoch, Jaromír (referee)
In this thesis, we study reweighting when estimating totals in survey sampling. The purpose of reweighting is to adjust the structure of the sample in order to comply with the structure of the population (with respect to given auxiliary variables). We sum up some known results for methods of the traditional desinbased approach, more attention is given to the modelbased approach. We generalize known asymptotic results in the modelbased theory to a wider class of weighted estimators. Further, we propose a consistent estimator of asymptotic variance, which takes into consideration weights used in estimator of the total. This is in contrast to usually recommended variance estimators derived from the designbased approach. Moreover, the estimator is robust againts particular model misspecifications. In a simulation study, we investigate how the proposed estimator behaves in comparison with variance estimators which are usually recommended in the literature or used in practice. 1


Testing equivalence and noninferiority
Rychterová, Nela ; Antoch, Jaromír (advisor) ; Omelka, Marek (referee)
This master thesis deals with topics related to the task whether customers are able to recognize a difference between products. First, testing of equivalence and noninferiority is discussed in detail. It is an important tool when verifying that two products are equivalent or that a new product is not substantially worse than a current product. Afterwards, Thurstone's approach is introduced as a way to evaluate the impact of a stimulus on human senses. Subsequently, using the previous chapters, there is a detailed discussion dealing with three standards wi dely used in practice in the case when someone needs to apply sensory evaluation to verify whether customers are able to recognize a difference between products. In particular, these are duotrio, triangle and paired comparison tests. There is a thorough explanation of their statistical base and the tests are compared accor ding to their power. Furthermore, an approach based on the Thurstone's theory is introduced as an alternative to the standard methods. Moreover, this thesis introduces Saaty's approach to the estimation of a priority vector, which is a useful tool to compare, to order or to choose the best one from n objects. We also introduce another approach to estimation of a priority vector which is based on Saaty's idea. 1


Neighborhood components analysis and machine learning
Hanousek, Jan ; Antoch, Jaromír (advisor) ; Maciak, Matúš (referee)
In this thesis we focus on the NCA algorithm, which is a modification of knearest neighbors algorithm. Following a brief introduction into classification algorithms we overview KNN algorithm, its strengths and flaws and what lead to the creation of the NCA. Then we discuss two of the most widely used mod ifications of NCA called Fast NCA and Kernel (fast) NCA, which implements the socalled kernel trick. Integral part of this thesis is also a proposed algo rithm based on KNN (/NCA) and Linear discriminant analysis titled TSKNN (/TSNCA), respectively. We conclude this thesis with a detailed study of two real life financial problems and compare all the algorithms introduced in this thesis based on the performance in these tasks. 1


Ratio estimators
Klyuchevskiy, Iakov ; Hlávka, Zdeněk (advisor) ; Antoch, Jaromír (referee)
The aim of the bachelor thesis is to estimate the incidence of fractures in women from 0 to 20 years in the Czech Republic. In the introductory chapter we will introduce the concept of incidence and show the statistical data that we will continue to work with. In the second and third chapters we define statistical models for estimating the incidence and also the unit estimation by which we estimate the incidence, we will examine its properties. In the fourth chapter, we will show the real data to estimate the incidence of fractures in women for each age category.


Transformation Models
Pejřimovský, Pavel ; Hušková, Marie (advisor) ; Antoch, Jaromír (referee)
This thesis deals with a finding ideal transformation which can model data well. We focus on transformations which we know up to a parametr. We need to estimate the parametr of the transformation. The main approach of study transformation is in linear regression and in nonparametric regression. In both cases we focus on estimating the transformation parametr and properties of this estimator such as consistency and asymptotic normality. We show in linear regression that the aprroach of least squares do not work properly. Instead of this we use a generalized moment method which can estimate parametr of transformation and also a regression coefficients. We show also a different solution for our problem in specific transformation called BoxCox. For this situation we make a simulation study for estimators and standard deviations. The standard deviation are obtained by bootstrap method. In nonparametric regression we use profile likelihood to estimate transformation parametr. We also construct an estimator of density of error terms. In both cases we know the asymptotic distribution.


Causes of Effects and Effects of Causes
Zemánková, Lucie ; Maciak, Matúš (advisor) ; Antoch, Jaromír (referee)
The thesis deals with an associative and causal relationship between two different random phenomena and presents basic statistical methods for investigation of these relationships. Firstly it focuses on demonstrating the association between phenomena and shows that finding a causal relation between phenomena requires appropriate randomization of the system or intervention in the system. After intervening in the system, it is no longer possible to observe all situations, socalled counterfactual observation, but the causal relationship can still be demonstrated using appropriate technical procedures and theoretical assumptions. The thesis further summarizes different ways of representation of causal structures, first by means of graphs, where basic methods of estimating the causal structure are presented, and later by structural equations that already capture the quantitative measure of causal relations.
