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Archimedean copulas
Vedyushenko, Anna ; Pešta, Michal (advisor) ; Omelka, Marek (referee)
The thesis deals with Archimedean copulas which are very popular nowadays due to easy construction and their appealing properties. At first it introduces the general definition of a copula and also shows its fundamental properties. After that the definition and the basic properties of an Archimedean copula are discussed. The paper also describes some of the commonly used families of Archi- medean copulas. Then several methods of parameter estimation for Archimedean copulas are shown. Finally, we make a study of two real datasets where the distri- bution of the data is estimated based on the procedures described in the thesis. 1
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Aggregation of dependent risks
Asipenka, Anna ; Mazurová, Lucie (advisor) ; Omelka, Marek (referee)
In this thesis we are interested in the calculation of economic capital for the to- tal loss which is the sum of partial dependent losses, whose dependence structure is described by Archimedean and hierarchical Archimedean copulas. Firstly, the concept of economic capital and the ways of its aggregation are introduced. Then the basic definitions and properties of copulas are listed, as well as the depen- dence measures. After that we work with definition and properties of Archimedean copulas and their simulation. We also mention the most popular families of Ar- chimedes copulas. Next, hierarchical Archimedean copulas are defined, as well as the algorithm for their sampling. Finally, we present methods for estimating the parameters of copulas and the recursive algorithm for estimating the hierarchical Archimedean copula structure. In the last chapter we perform simulation studies of selected models using hierarchical Archimedes copulas. 1
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Archimedean copulas
Vedyushenko, Anna ; Pešta, Michal (advisor) ; Omelka, Marek (referee)
The thesis deals with Archimedean copulas which are very popular nowadays due to easy construction and their appealing properties. At first it introduces the general definition of a copula and also shows its fundamental properties. After that the definition and the basic properties of an Archimedean copula are discussed. The paper also describes some of the commonly used families of Archi- medean copulas. Then several methods of parameter estimation for Archimedean copulas are shown. Finally, we make a study of two real datasets where the distri- bution of the data is estimated based on the procedures described in the thesis. 1
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A note on the use of copulas in chance-constrained programming
Houda, Michal
In this paper we are concentrated on a problem of linear chanceconstrained programming where the constraint matrix is considered random with a known distribution of the matrix rows. The rows are not considered to be independent; instead, we make use of the copula notion to describe the dependence of the matrix rows. In particular, the distribution of the rows is driven by so-called Archimedean class of copulas. We provide a review of very basic properties of Archimedean copulas and describe how they can be used to transform the stochastic programming problem into a deterministic problem of second-order cone programming. Also the question of convexity of the problem is explored and importance of the selected class of copulas is commented. At the end of the paper, we provide a simple example to illustrate the concept used.
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Important Markov-Chain Properties of (1,lambda)-ES Linear Optimization Models
Chotard, A. ; Holeňa, Martin
Several recent publications investigated Markov-chain modelling of linear optimization by a (1,lambda)-ES, considering both unconstrained and linearly constrained optimization, and both constant and varying step size. All of them assume normality of the involved random steps. This is a very strong and specific assumption. The objective of our contribution is to show that in the constant step size case, valuable properties of the Markov chain can be obtained even for steps with substantially more general distributions. Several results that have been previously proved using the normality assumption are proved here in a more general way without that assumption. Finally, the decomposition of a multidimensional distribution into its marginals and the copula combining them is applied to the new distributional assumptions, particular attention being paid to distributions with Archimedean copulas.
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Modelling natural catastrophes in insurance
Varvařovský, Václav ; Zimmermann, Pavel (advisor) ; Justová, Iva (referee)
Quantification of risks is one of the pillars of the contemporary insurance industry. Natural catastrophes and their modelling represents one of the most important areas of non-life insurance in the Czech Republic. One of the key inputs of catastrophe models is a spatial dependence structure in the portfolio of an insurance company. Copulas represents a more general view on dependence structures and broaden the classical approach, which is implicitly using the dependence structure of a multivariate normal distribution. The goal of this work, with respect to absence of comprehensive monographs in the Czech Republic, is to provide a theoretical basis for use of copulas. It focuses on general properties of copulas and specifics of two most commonly used families of copulas -- Archimedean and elliptical. The other goal is to quantify difference between the given copula and the classical approach, which uses dependency structure of a multivariate normal distribution, in modelled flood losses in the Czech Republic. Results are largely dependent on scale of losses in individual areas. If the areas have approximately a "tower" structure (i.e., one area significantly outweighs others), the effect of a change in the dependency structure compared to the classical approach is between 5-10% (up and down depending on a copula) at 99.5 percentile of original losses (a return period of once in 200 years). In case that all areas are approximately similarly distributed the difference, owing to the dependency structure, can be up to 30%, which means rather an important difference when buying the most common form of reinsurance -- an excess of loss treaty. The classical approach has an indisputable advantage in its simplicity with which data can be generated. In spite of having a simple form, it is not so simple to generate Archimedean copulas for a growing number of dimensions. For a higher number of dimensions the complexity of data generation greatly increases. For above mentioned reasons it is worth considering whether conditions of 2 similarly distributed variables and not too high dimensionality are fulfilled, before general forms of dependence are applied.
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