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Basic Multivariate Distributions
Sýkorová, Sabina ; Kulich, Michal (advisor) ; Hurt, Jan (referee)
The thesis deals with the basic discrete and continuous multivariate distributions, which play an important role in statistical analyses of models in applied fields. It focuses mainly on the derivation of these distributions using various techniques by which univariate distributions are generalized to higher dimensions. At the beginning of the thesis the multivariate normal distribution is defined, than it deals with distributions that are derived by direct generalization of univariate distributions. These are multivariate log-normal, multivariate Student's, multivariate Pareto, Dirichlet, and multinomial distributions. Furthermore it describes a common components method by which a multivariate Poisson distribution and a multivariate gamma distribution are derived. In the last chapter we introduce a multivariate exponential distribution derived by a stochastic generalization technique.
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Selected problems and methods in multivariate data analysis
Goduľová, Lenka ; Zichová, Jitka (advisor) ; Hurt, Jan (referee)
Title: Selected problems and methods in multivariate data analysis Author: Lenka Goduľová Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Jitka Zichová, Dr. Abstract: The bachelor thesis deals with processing multidimensional data. The task was to apply selected methods on financial data. The thesis is composed of the theoretical section and the analysis of a particular database. The first four chapters deal with basic relations and definitions concerning random vector and variable, multidimensional data and the independence test in a contingency table. The following section is devoted to defining the particular methods selected: cluster analysis and discriminant analysis. In the practical section these methods are applied to a database of clients of a German bank. Keywords: random vector, multivariate distribution, multivariate random variable, contingency table, cluster analysis, discriminant analysis.
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Spherically symmetric measures
Ranošová, Hedvika ; Nagy, Stanislav (advisor) ; Dvořák, Jiří (referee)
A probability distribution is called spherically symmetric if it is invariant with respect to rotations about the origin. This class includes the multivariate standard normal distribution, a multivariate extension of the t-distribution and uniform distribu- tions inside the unit ball or the unit sphere surface. The first part of the thesis summarizes the basic properties of spherically symmetric distributions such as the form of their char- acteristic function and provides expressions for their moments and the density function. It turns out that spherically symmetric distributions are fully characterized by the dis- tribution of their Euclidean norm or by any of their univariate marginal distributions. As any marginal distribution of a spherically symmetric distribution is also spherically symmetric, the aim of the second part of this thesis is to study the inverse relationship using fractional calculus. For a given n-dimensional spherically symmetric distribution we solve the problem of deciding whether there is a spherically symmetric distribution in higher dimensions whose n-dimensional marginal is as given. 1
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Multivariate Pareto distribution
Novytskyi, Oleksandr ; Mazurová, Lucie (advisor) ; Pešta, Michal (referee)
Title: Multivariate Pareto distribution Author: Oleksandr Novytskyi Department: Department of Probability and Mathematical Statistics (305. 32- KPMS) Supervisor: RNDr. Lucie Mazurová, Ph.D., Department of Probability and Mathematical Statistics (305. 32-KPMS) Abstract: This bachelor thesis focuses on three methods of constructing multiva- riate Pareto distribution, i.e. multivariate distribution, where marginal distributi- ons are univariate Pareto distributions. We provide survival and density functions for these models, which are used for the numerical studies and valuation of insu- rance product, specifically a yearly life annuity paid to each insured in the group, whose remaining life time is given by the multivariate Pareto distribution. Keywords: multivariate distribution, Pareto distribution, survival function, density, life annuity.
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Selected problems and methods in multivariate data analysis
Goduľová, Lenka ; Zichová, Jitka (advisor) ; Hurt, Jan (referee)
Title: Selected problems and methods in multivariate data analysis Author: Lenka Goduľová Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Jitka Zichová, Dr. Abstract: The bachelor thesis deals with processing multidimensional data. The task was to apply selected methods on financial data. The thesis is composed of the theoretical section and the analysis of a particular database. The first four chapters deal with basic relations and definitions concerning random vector and variable, multidimensional data and the independence test in a contingency table. The following section is devoted to defining the particular methods selected: cluster analysis and discriminant analysis. In the practical section these methods are applied to a database of clients of a German bank. Keywords: random vector, multivariate distribution, multivariate random variable, contingency table, cluster analysis, discriminant analysis.
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Basic Multivariate Distributions
Sýkorová, Sabina ; Kulich, Michal (advisor) ; Hurt, Jan (referee)
The thesis deals with the basic discrete and continuous multivariate distributions, which play an important role in statistical analyses of models in applied fields. It focuses mainly on the derivation of these distributions using various techniques by which univariate distributions are generalized to higher dimensions. At the beginning of the thesis the multivariate normal distribution is defined, than it deals with distributions that are derived by direct generalization of univariate distributions. These are multivariate log-normal, multivariate Student's, multivariate Pareto, Dirichlet, and multinomial distributions. Furthermore it describes a common components method by which a multivariate Poisson distribution and a multivariate gamma distribution are derived. In the last chapter we introduce a multivariate exponential distribution derived by a stochastic generalization technique.
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