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Occupation of a set time of random walks
Janoušek, Jan ; Hlubinka, Daniel (advisor) ; Karafiátová, Iva (referee)
This bachelor's thesis aims to analyse random walks, emphasizing symmetric random walks. We focus mainly on the occupation of a set times. In this thesis, we find the distribution of maxima and minima, the probability of return to the origin, and the probability of first return to the origin. Then we see the arcsine laws and the probability of spending a given amount of time on the positive and the negative side. We generalize this not only for the positive and negative sides but for any given interval. At the end of the thesis, we create statistical tests based on the theoretical distribution derived in the thesis. 1
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Time reversibility of random process
Paclík, Ondřej ; Hlubinka, Daniel (advisor) ; Hudecová, Šárka (referee)
Random processes can be used to describe the evolution of a real systems over time. Discrete-time Markov chains are random processes that meet special assumptions, but they still have a lot of practical applications. Some chains have the property that it is impossible to tell if they are being observed when the passage of time is reversed. We call such chains time reversible. In this paper, we define a time reversible Markov chain with discrete time, we show how it can be verified that a given chain is time reversible, and we introduce basic properties and examples of time reversible chains. At the same time, we apply the knowledge of time reversibility to the problem of finding the stationary distribution of specific Markov chains. 1
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Skorokhod's representation theorem
Paulová, Nikol ; Nagy, Stanislav (advisor) ; Hlubinka, Daniel (referee)
We know that almost sure convergence of random variables implies their convergence in distribution. Are there any conditions that would allow us to obtain al- most sure convergence from convergence in distribution? The Skorokhod representation theorem answers this question. We can find representations of the weakly convergent random variables such that they converge almost surely. First, we introduce the needed definitions and lemmata. The main focus of the second chapter is the Skorokhod repre- sentation theorem on the real numbers, its proof and some auxiliary assertions are given. In the final third chapter, we deal with the applications of the theorem to prove some well known and commonly used theorems and to prove some less known theorems. 1
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Geometric distribution and its multivariate version
Pavlovičová, Diana ; Hlubinka, Daniel (advisor) ; Pawlasová, Kateřina (referee)
In this work we will discuss the basics of a multivariate geometric distribution, especially its two-dimensional version. First of all, we establish a fundamental definition in which we consider two types of failures. Next, we compute some of its properties. We then focus on a different version of the two-dimensional case which we obtain by conditioning and for which we again compute its properties. We extend this approach to the case where we consider three types of failures. We further generalize the obtained results for the case of a multivariate negative binomial distribution. Lastly, we focus on the estimates of the parameters of the fundamental two-dimensional version of the multivariate geometric distribution and present a simple simulation in which we demonstrate the accuracy of the obtained estimates. 1
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Cox model with interval-censored data
Štarmanová, Petra ; Komárek, Arnošt (advisor) ; Hlubinka, Daniel (referee)
Survival analysis typically deals with censored data. This thesis focuses on interval- censored data, which are common in medical studies. We present regression models for analysing interval-censored data with emphasis on semiparametric models. We study the models of Finkelstein and Farrington in depth and show their use on real data. The properties of both models are explored in a simulation study. 1
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Regression goodness-of-fit criteria according to dependent variable type
Šimsa, Filip ; Hanzák, Tomáš (advisor) ; Hlubinka, Daniel (referee)
This work is devoted to the description of linear, logistic, ordinal and multinominal regression models and interpretation of its parameters. Then it introduces a variety of quality indicators of mathematical models and the re- lations between them. It focuses mainly on the Gini coefficient and the coefficient of determination R2 . The first mentioned is established by modifying the Lorenz curve for ordinal and continuous variables and by comparing the estimated proba- bilities for nominal variable. The coefficient of determination R2 is newly defined for the nominal variable and is examined its relationship with Gini coefficient. As- suming normally distributed scores and errors of the model is numerically derived the relation between the Gini coefficient and the coefficient of determiantion for different distribution of continuous dependent variable. Theoretical calculations and definitions are illustrated on two real data sets. 1
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The Depth of Functional Data.
Nagy, Stanislav ; Hlubinka, Daniel (advisor) ; Omelka, Marek (referee)
The depth function (functional) is a modern nonparametric statistical analysis tool for (finite-dimensional) data with lots of practical applications. In the present work we focus on the possibilities of the extension of the depth concept onto a functional data case. In the case of finite-dimensional functional data the isomorphism between the functional space and the finite-dimensional Euclidean space will be utilized in order to introduce the induced functional data depths. A theorem about induced depths' properties will be proven and on several examples the possibilities and restraints of it's practical applications will be shown. Moreover, we describe and demonstrate the advantages and disadvantages of the established depth functionals used in the literature (Fraiman-Muniz depths and band depths). In order to facilitate the outcoming drawbacks of known depths, we propose new, K-band depth based on the inference extension from continuous to smooth functions. Several important properties of the K-band depth will be derived. On a final supervised classification simulation study the reasonability of practical use of the new approach will be shown. As a conclusion, the computational complexity of all presented depth functionals will be compared.
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Sentence length distribution
Kašpar, Martin ; Hlávka, Zdeněk (advisor) ; Hlubinka, Daniel (referee)
In the present work we study whether it is possible to describe the lengths of sentences of a prosaic text by a probability distribution. We focus on negative binomial, lognormal and Sichel distributions and their comparison. We study Sichel distribution in detail, because it was introduced as distribution for description of bibliometric data. We also investigate estimation of parameters of all three distributions and then use the theoretical results on specific data (a few texts in Czech and English). Finally, we test the accuracy of the distributions and estimated parameters, using the results given in this work. 1
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