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Applications of random walk in queueing theory
Uhliar, Miroslav ; Hlubinka, Daniel (advisor) ; Antoch, Jaromír (referee)
The bachelor thesis "Applications of random walk in queueing theory" is about an approach functioning queueing theory, in other words, system where the costumers are served by a server. We describe kinds of queues, services, with different number of servers. In the first chapter, the attention is devoted searching for stationary distributions. Subsequently, in the second chapter, there is described the relation random walk with waiting time for service. We use for that Lindley process. It includes also the most important statement of all thesis describing that relation. In the section "Chosen problems and their solutions" we may find the application of the theory.
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Functional data and their principal components analysis
Kasanický, Ivan ; Hlubinka, Daniel (advisor) ; Hušková, Marie (referee)
Presented thesis deals with analysis of functional data. In the first part, problem which arises because of only finite possible numbers of observations is discussed. This problem is solved using representation by basis functions with emphasis on B-splines basis. The second part is focused on functional principal component analysis that could be understood as a natural extension of a multivariate case or as an application of Karhunen-Lo`eve expansion , which is based on Mercer's theorem. Estimations of principal components together with rates of convergence are mentioned too. Practical computation of principal components is mentioned in the last chapter.
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Fisherovo-Binghamovo rozdělení
Malá, Olivia Caroline ; Hlávka, Zdeněk (advisor) ; Hlubinka, Daniel (referee)
This thesis is an introduction into directional statistics, a subdiscipline of statistics that occupies itself with directional data. Because of the special structure of the sample spaces, which are n-dimensional hyperspheres, the statis- tical theory has to be adjusted. We start on the circle, where we define the circular random variable (also called random angle) together with its characterizations, and continue with studying estimators of its parameters. Subsequently, we generalize the results to the n- dimensional case. Further follows an overview of the Fisher-Bingham family of probability distributions with a more detailed presentation of the von Mises dis- tribution as a representative of the family on the circle. 1
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Level Sets of Multivariate Density Functions and their Estimates
Kubetta, Adam ; Hlubinka, Daniel (advisor) ; Zichová, Jitka (referee)
A level set of a function is defined as the region, where the function gets over the specified level. A level set of the probability density function can be considered an alternative to the traditional confidence region because on certain conditions the level set covers the region with minimal volume over all regions with a given confidence level. The benefits of using level sets arise in situations where, for example, the given random variables are multimodal or the given random vectors have strongly correlated components. This thesis describes estimates of the level set by means of a so called plug-in method, which first estimates density from the data set and then specifies the level set from the estimated density. In addition, explicit direct methods are also studied, such as algorithms based on support vectors or dyadic decision trees. Special attention is paid to the nonparametric probability density estimates, which form an essential tool for plug-in estimates. Namely, the second chapter describes histograms, averaged shifted histograms, kernel density estimates and its generalization. A new technique transforming kernel supports is proposed to avoid the so called boundary effect in multidimensional data domains. Ultimately, all methods are implemented in Mathematica and compared on financial data sets.
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The identification function for the convergence in probability with an application in the estimation theory
Kříž, Pavel ; Štěpán, Josef (advisor) ; Hlubinka, Daniel (referee)
In the present work we introduce the concept of probability limit identification function (PLIF) as it is done in [6]. This function identifies almost surely the value of the probability limit of a sequence of random variables on the basis of one realization of the sequence. According to the same article we show the construction of PLIF for real valued random variables from the special PLIF for 0-1 valued random variables. Following the method described in [8] we prove the existence of the universal PLIF for real valued random variables under the continuum hypothesis. Next we show that there are no borel measurable special PLIFs for 0-1 valued random variables (as well as PLIFs for real valued random variables). We use the proof that is published in [2]. Then we extend the construction of PLIF from R to any separable metrizable topological space. This PLIF may be used e.g. for creating functional representations of stochastic integrals and weak solutions of stochastic differential equations.
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Maximum likelihood methods; selected problems
Chlubnová, Tereza ; Hlubinka, Daniel (advisor) ; Hlávka, Zdeněk (referee)
Maximum likelihood estimation is one of statistical methods for estimating an unknown parameter. It is often used because of a simple calculation of the estimator and also for characteristics of this estimator, which the method provides under some conditions. In the thesis we prove a consistence of the estimator under conditions of regularity and uniqueness of the root of the likelihood equation. If we add other assumptions we show its asymptotic normality and we expand this result from the one-dimensional parameter to the multi-dimensional parameter. The main result of the thesis lies in exercises, in which we cannot express the maximum likelihood estimator in general, but we can show its existence, uniqueness and asymptotic normality. Moreover we demonstrate the utilization of asymptotic normality of the estimator for asymptotic hypothesis tests and confidence intervals of the parameter. Powered by TCPDF (www.tcpdf.org)
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