|
|
Quantiles for directional data
Fedor, Jakub ; Nagy, Stanislav (advisor) ; Hlubinka, Daniel (referee)
In this thesis we study a special type of multidimentional data - directional data. The main part of this thesis consists of defining and comparing different ways of order- ing directional data. The most important functions used for ordering directional data presented in this thesis are angular depths. We will describe importnant properties of angular depths and we will discuss wheter each angular depth satisfies the formulated desirable properties. Using previously defined angular depths and median we show two ways of drawing the circular version of a boxplot. 1
|
|
|
α-symmetric measures
Ranošová, Hedvika ; Nagy, Stanislav (advisor) ; Hlubinka, Daniel (referee)
Spherically symmetric measures in Rn are rotationally invariant, indicating that their characteristic functions can be written as a composition of the Euclidean norm with a univariate function. If we replace the Euclidean norm with an ℓα norm, the resulting distributions are known as α-symmetric. This thesis aims to provide a general description of α-symmetric measures and explore various non-trivial examples. The existence of α- symmetric measures for a given α and dimension n ∈ N is discussed, along with the connection between the existence of α-symmetric measures and isometric embedding into Lp spaces through strictly stable distributions. One of the main properties explored in this thesis is the relationship between moments of non-integer order and α-symmetry in distributions. Additionally, several sufficient conditions for the existence and the form of α-symmetric measures are described. In the final chapter, a further generalization of α-symmetric distributions toward quasi-norms is discussed, along with the properties of the resulting concept of pseudo-isotropy. 1
|
|
|
James-Stein Estimator
Novotný, Vojtěch ; Maciak, Matúš (advisor) ; Nagy, Stanislav (referee)
In this thesis, we will introduce the James-Stein estimator, we will study its properties and compare them with other estimation methods. We will explain, what is admissibility of an estimator and figure out if our estimators are admissable. We will introduce the Bayes estimators and the empirical Bayes estimators. Furthermore, we will analyse how their properties can be examined differently. Finally, we will perform a simulation study and we will compare the quality of estimations on its results and see if they follow the explained theory. Using this, we will try to decide when is using the James-Stein estimator appropriate. 1
|
|
|
Delta method and its generalizations
Pavlech, Ján ; Omelka, Marek (advisor) ; Nagy, Stanislav (referee)
The goals of this thesis are various generalizations of the classical delta theorem, in which the advantage is that we can separately investigate the analytical properties of transformation of the estimate, and independently, we can deal with asymptotic properties of the original estimate. When working with Euclidean spaces, we generalize the delta theorem for the case that partial derivatives are not continuous or they are equal to zero. When working with general normed linear spaces, we first examine Hadamard- differentiability, while formulating and proving equivalence with Fréchet-differentiability, under proper assumptions. We demonstrate the functional delta theorem on known results for empirical quantiles and median absolute deviation in the case of a random sample, together with our own result for the interquartile range and empirical quantiles in the case of AR(d) sequence. We also show why the functional delta theorem is not usable for moment estimators. In the last part, we examine the Hadamard-differentiability of a copula functional and its application to the derivation of the asymptotic distribution of the empirical copula. 1
|
|
|
Simplicial depth
Mendroš, Erik ; Nagy, Stanislav (advisor) ; Hlubinka, Daniel (referee)
Depth functions play a crucial role in nonparametric statistics by generalizing orderings, ranks, and quantiles to multivariate data. In our thesis, we provide a comprehensive study of the classical and revised definitions of simplicial depth function, accompanied by detailed and illustrated proofs of some of their proper- ties. Our research also addresses some issues in previous publications and explores potential expansions of those concepts. In the final part of the thesis, we reveal an intriguing connection between simplicial depth and Sylvester's four-point prob- lem, which may have implications for future advancements in this field. 1
|
|
|
Theoretical and empirical quantiles and their use for prediction interval construction
Šimičák, Jakub ; Maciak, Matúš (advisor) ; Nagy, Stanislav (referee)
The purpose of the bachelor thesis is to introduce the reader to two approaches to the construction of prediction intervals. The first procedure assumes a probabilistic model and leads to a frequentist prediction interval that uses the relevant theoretical quantiles of probability distributions. The second procedure assumes no probabilistic model and leads to a conformal prediction interval that uses empirical quantiles of the relevant random selection. In the course of the paper, both approaches will be derived in general terms and then illustrated with concrete examples. The thesis also includes a simulation study comparing the empirical coverage of frequentist and conformal prediction inter- vals for random selections from different distributions. 1
|
|
|
Anderson's theorem
Bočinec, Filip ; Nagy, Stanislav (advisor) ; Lachout, Petr (referee)
In this thesis we explore a theorem from real analysis and geometry called Anderson's theorem. It concerns an integral inequality for symmetric quasi-concave functions, where the integration is done over a symmetric convex set. A thorough proof of Anderson's theorem is given. In addition, we investigate cases in which equality or strict inequality occurs. While studying this topic, we come across some problems in published papers and we try to clarify them. Furthermore, we explore possible extensions of the theorem. In particular, results involving group invariance and theory of s-concave functions are mentioned. As outlined in the final part of the thesis, Anderson's theorem is a useful and widely used tool in multivariate statistics. 1
|
|
|
Regression depth and related methods
Dočekalová, Denisa ; Nagy, Stanislav (advisor) ; Omelka, Marek (referee)
While the halfspace depth has gained more and more popularity in the recent years as a robust estimator of the mean, regression depth, despite being based on a similar concept, is still a relatively unknown method. The main goal of this paper was therefore to introduce the concept of robust depth to the reader, illustrate its geometric interpre- tation, and provide at least a basic overview of the findings that occurred within the individual researches. Finally, a small simulation study was conducted comparing the de- epest regression method with other selected methods commonly used in practice, namely the method of least absolute deviations and ordinary least squares method. 1
|
|
|
EM algorithm for truncated Gaussian mixtures
Nguyenová, Adéla ; Dvořák, Jiří (advisor) ; Nagy, Stanislav (referee)
The expectation-maximization iterative algorithm is widely used in parameter estimation when dealing with missing information. Such a situation can naturally arise when we observe the data of our interest on a bounded observation window. This thesis focuses on the application of the EM algorithm for truncated Gaussian mixtures and compares the proposed algorithm with the approach in a previously published article, see Lee and Scott [2012], where it uses a heuristic simplification and is not sufficiently supported mathematically. We also compare the behavior of the proposed algorithm with the procedure from the article in a series of simulated experiments, as well as in analyzing a real dataset. We also provide Python implementation of the EM algorithm for truncated Gaussian mixtures.
|
|
|
Stein's method for normal approximation of random variables
Strnad, Martin ; Kříž, Pavel (advisor) ; Nagy, Stanislav (referee)
The Stein's method is a collection of probabilistic techniques for answering the ques- tion as to how far the probability distributions of random variables are from each other. This thesis only concerns the basics of the approach. We use the Kolmogorov distance and the total variation distance to formalize the concept of the distance between mea- sures. We focus on the normal distribution for which we first find a suitable differential operator, often called Stein operator, that bears much information. Not only does it charactize the Gaussian measure, it also gives us a means to quantify the distance from another random variable's distribution. Finally, we apply the method to prove the clas- sic Berry-Esseen inequality for a sum of independent and identically distributed random variables. 1
|
guest ::
