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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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L1 Regression
Čelikovská, Klára ; Maciak, Matúš (advisor) ; Hlubinka, Daniel (referee)
This thesis is focused on the L1 regression, a possible alternative to the ordinary least squares regression. L1 regression replaces the least squares estimation with the least absolute deviations estimation, thus generalizing the sample median in the linear regres- sion model. Unlike the ordinary least squares regression, L1 regression enables loosening of certain assumptions and leads to more robust estimates. Fundamental theoretical re- sults, including the asymptotic distribution of regression coefficient estimates, hypothesis testing, confidence intervals and confidence regions, are derived. This method is then compared to the ordinary least squares regression in a simulation study, with a focus on heavy-tailed distributions and the possible presence of outlying observations. 1
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Essential problems of random walks
Michálek, Matěj ; Hlubinka, Daniel (advisor) ; Pawlas, Zbyněk (referee)
In this paper, we cover some essential problems of (simple) random walks in one, two and three dimensions. At the begining, we work only in one dimension. We find the probability of a position on a line at particular time. Then we study returns to origin and examine if return to origin is certain. Also, we look into a theorem called the arc sine law. Furthermore, we generalise some of those problems into two and three dimensions. We investigate a probability of a position in time and space and returns to origin. 1
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Actuarial and Exposure-based Models for Hail Peril
Drobuliak, Matúš ; Pešta, Michal (advisor) ; Hlubinka, Daniel (referee)
Title: Actuarial and Exposure-based Models for Hail Peril Author: Bc. Matúš Drobuliak Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Michal Pešta, Ph.D., Department of Probability and Mathe- matical Statistics Abstract: This thesis covers an introduction to catastrophe modelling and focuses on statistical methods for extreme events. This includes methods of estimating parameters of claim distribution with a focus on probability weighted moments estimation technique. Furthermore, times series modelling, skew t-distribution, and two model clustering techniques are examined as well. This is later utilised in the practical application part of this thesis, which uses real data provided by an insurance company operating in the Czech Republic. Probability distribution fitting of large claims caused by hailstorms and Monte Carlo simulation of future losses are demonstrated later. Keywords: Catastrophe modelling, Hail peril, Probability weighted moments, Extreme events, ARMA-GARCH, Monte Carlo simulation iii
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Halfspace median
Říha, Adam ; Nagy, Stanislav (advisor) ; Hlubinka, Daniel (referee)
In this thesis we introduce the halfspace median, which is one of the possibilities how to extend the classical median from a one-dimensional space to spaces with several dimensions. Firstly we deal with the halfspace depth, which is a function that assigns to each point the minimum probability of a halfspace that contains it. Then we define the halfspace median and show its existence. Partially, we also deal with special types of symmetry measures for convex sets and random vectors and what follows from them, such as when the median and the center of symmetry are the same point. We also study the boundaries that, under certain assumptions, enclose the depth. We state sufficient conditions for acquiring the halfspace median, which are determined by the so-called ray basis theorem. Finally we look at the similarities of this topic with convex geometry.
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Love-Young Inequality and Its Consequences
Sýkora, Adam ; Čoupek, Petr (advisor) ; Hlubinka, Daniel (referee)
This thesis is focused on proving the Love-Young inequality and clarifying the manner in which it relates to a fractional Brownian motion. To begin with, several estimates alongside the concept of p-variation of a func- tion are presented. The connection between functions of finite p-variation and regulated functions is then highlighted and used to prove the aforementioned Love-Young inequality. Deficiency of the pathwise approach to stochastic in- tegration is recognised and later discussed amongst the properties of fractional Brownian motions. This constitutes the main application of the featured theory which is the integration with respect to irregular functions. 1
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