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Selected topics of random walks
Filipová, Anna ; Hlubinka, Daniel (advisor) ; Beneš, Viktor (referee)
The theme of this thesis are symmetric random walks. We define different types of paths and prove the reflection principle. Then, based on the paths, we define random walks. The thesis also deals with probabilities of returns to the origin and first returns to the origin, further with probabilities of number of changes of sign or returns to the origin up to a certain time. We also define the maximum of the random walk and the first passage through a certain point. In the second chapter, we solve several problems, which form the proofs of some theorems from the first chapter or complement the first chapter in a different way. For example, we prove geometrically that the number of paths of one type equals the number of paths of another type or we compute the probability that there occurs a certain number of changes of sign up to a given time.
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Depth of variance matrices
Brabenec, Tomáš ; Nagy, Stanislav (advisor) ; Hlubinka, Daniel (referee)
The scatter halfspace depth is a quite recently established concept which extends the idea of the location halfspace depth for positive definite matrices. It provides an interest- ing insight into the problem of suitability quantification of a matrix for the description of the covariance structure of the multivariate distribution. The thesis focuses on the investigation of theoretical properties of the depth for both general and more specific probability distributions which can be used for data analysis. It turns out that the es- timators of scatter parameters based on the empirical scatter depth are quite effective even under relatively weak assumptions. These estimators are useful especially for dealing with a sample containing outliers or contaminating observations. 1
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Multivariate distributions in Cartesian, polar and directional coordinates
Bečková, Magdaléna ; Hlubinka, Daniel (advisor) ; Pawlas, Zbyněk (referee)
The thesis focuses on the distributions of random vectors in Cartesian, polar and directional coordinates. In the thesis we derive formulas for probability density func- tions of two-dimensional vectors in polar and directional coordinates, three-dimensional vectors in spherical and directional coordinates and n-dimensional vectors in spherical coordinates. These formulas are shown on several examples of normal and uniform distri- butions. Finally, the thesis discusses differences between the probability density functions in particular coordinates systems. 1
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Probability distribution of functional random variables
Dolník, Viktor ; Hlubinka, Daniel (advisor) ; Hlávka, Zdeněk (referee)
We describe basic notions of functional random elements and the space of functions L2 [0, 1]. We discuss the non-existence of a probability density functional and the re- quirements for integrating in a functional space. In Chapter 2, we define distribution functionals and introduce a goodness-of-fit test which utilises them. The concept of char- acteristic functionals follows in Chapter 3, along with the latest test for Gaussianity of functional random elements. We conclude the chapter with our own new goodness-of- fit test, where we prove the distribution of its test statistic under the alternative, then under the null hypothesis, and lastly the distribution of the bootstrapped test statistic. Finally, we illustrate the theory on a simulation study of the empirical significance level and power of the goodness-of-fit tests. 1
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Selected properties of bivariate and multivariate random walks.
Nguyen, Huy Quang ; Hlubinka, Daniel (advisor) ; Kříž, Pavel (referee)
This thesis deals with random walks with emphasis on multivariate random walks. We focus mainly on return of the random walk to the origin in two dimensions. Some results are generalized in any dimension. Specifically we discuss the probability of return to the origin, probability of the first return to the origin and expected time of the first return. In the thesis we also find the arcsine laws and short simulation study focused on multivariate version of this topic. 1
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Functional data analysis
Jurica, Tomáš ; Hlávka, Zdeněk (advisor) ; Hlubinka, Daniel (referee)
The aim of the master thesis is to review of reconstruction techniques of func- tional data and existing one-way functional ANOVA (FANOVA) tests. Specif- ically, the work deals with L2 -norm based and F-type mean functions equality tests, L2 -norm based covariance functions equality tests and tests for distri- bution equality. Furthermore, for each type of the test, it is introduced test based on reconstructed functional data, using orthornormal basis functions of L2 space. Finally, simulation study was conducted for comparing properties of tests using orthonormal basis representation of functional data and tests applied on non-reconstructed data. 1
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Multivariate random walk model for multiple players games
Pavlech, Ján ; Hlubinka, Daniel (advisor) ; Večeř, Jan (referee)
The goal of this bachelor's thesis is to analyse a game of three players, as a multiva- riate random walk. Specifically, its probability distribution from a purely combinatoric approach, but also through generating functions and the inverse formula. We will exa- mine in detail the basic properties in a few simpler models: regular rotation of players who are equally skilled, regular rotation of players who are not equally skilled, and irregular rotation of players who are not equally skilled. We will also focus on the fairness of the game, return to its origin, and distribution of maximum achieved during the game. In the last chapter, we will inspect more closely some basic simulations of progress of the game. 1
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Generalization of convex functions
Bessisso, Samir ; Lachout, Petr (advisor) ; Hlubinka, Daniel (referee)
Convex functions have range of useful properties that can be well utilized in mathe- matical optimization. For instance, their local minima is also global minima, they have convex lower level sets and if differentiable, their stationary point is also the point of global minima. For differentiable convex functions gradient methods and Karush-Kuhn-Tucker conditions can be effectively applied. On the other hand, the assumption of convexity is rather restrictive and not necessary for some of their desired properties. Theme of this thesis are convex functions and their generalizations, namely quasiconvex and K-convex functions, invex functions are also marginally mentioned. This thesis gathers knowledge about convex, quasiconvex and K-convex functions that can be used in mathematical optimization and ilustrates it on examples. 1
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Parameter Estimation in Stochastic Differential Equations
Pacák, Daniel ; Maslowski, Bohdan (advisor) ; Hlubinka, Daniel (referee)
In the Thesis the problem of estimating an unknown parameter in a stochastic dif- ferential equation is studied. Linear equations with Volterra process as the source of noise are considered. Firstly, the properties of Volterra processes and the properties of stochastic integral with respect to a Volterra process are presented. Secondly, the prop- erties of the solution to the equation under consideration are discussed. This includes the existence of the strictly stationary solution, the properties of such solution and ergodic results. These results are then generalized to equations with a mixed noise. Ergodic results are used to derive strongly consistent estimators of the unknown parameter. 1
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Zonoids of measures and their applications
Hendrych, František ; Nagy, Stanislav (advisor) ; Hlubinka, Daniel (referee)
In the present thesis we are concerned with special convex sets called zonoids. Zonoids are sets that are possible to be expressed as a limit case of a finite sum of line segments. They have found applications in geometry or functional analysis. The subject of our study are mainly the properties of a mapping that to a properly integrable Borel measure assigns a zonoid constructed from that measure. That mapping has an array of interesting properties. It turns out, however, that it is not injective. A solution to this problem is first to apply a suitable transform to the measure, and then to construct a zonoid of the transformed measure. The resulting set is called the lift zonoid of a measure. The mapping that to measure assigns its lift zonoid can be shown to be injective. As we outline in the final part of the thesis, lift zonoids of measures find important applications in multivariate statistics. 1
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