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Statistics of extremes
Fusek, Michal ; Neubauer, Jiří (referee) ; Michálek, Jaroslav (advisor)
The thesis deals with extreme value distributions. The theoretical part is devoted to the basics of extreme value theory and to the characterization of extreme value distributions. There is the limit theorem for distributions of the maximum formulated and characteristics of the extreme value distributions deduced. There are parameter estimates for Weibull, lognormal and exponential distributions inferred using method of maximum likelihood and method of moments. There is also the theory of censored samples described. The practical part is devoted to statistical analysis of rainfall.
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Non-parametric estimation of parameters of extreme value distribution
Blachut, Vít ; Popela, Pavel (referee) ; Michálek, Jaroslav (advisor)
The concern of this diploma thesis is extreme value distributions. The first part formulates and proves the limit theorem for distribution of maximum. Further there are described basic properties of class of extreme value distributions. The key role of this thesis is on non-parametric estimations of extreme value index. Primarily, Hill and moment estimator are derived, for which is, based on the results of mathematical analysis, suggested an alternative choice of optimal sample fraction using a bootstrap based method. The estimators of extreme value index are compared based on simulations from proper chosen distributions, being close to distribution of given rain-fall data series. This time series is recommended a suitable estimator and suggested choice of optimal sample fraction, which belongs to the most difficult task in the area of extreme value theory.
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Extreme Value Distribution Parameter Estimation and its Application
Holešovský, Jan ; Picek,, Jan (referee) ; Antoch,, Jaromír (referee) ; Michálek, Jaroslav (advisor)
The thesis is focused on extreme value theory and its applications. Initially, extreme value distribution is introduced and its properties are discussed. At this basis are described two models mostly used for an extreme value analysis, i.e. the block maxima model and the Pareto-distribution threshold model. The first one takes advantage in its robustness, however recently the threshold model is mostly preferred. Although the threshold choice strongly affects estimation quality of the model, an optimal threshold selection still belongs to unsolved issues of this approach. Therefore, the thesis is focused on techniques for proper threshold identification, mainly on adaptive methods suitable for the use in practice. For this purpose a simulation study was performed and acquired knowledge was applied for analysis of precipitation records from South-Moravian region. Further on, the thesis also deals with extreme value estimation within a stationary series framework. Usually, an observed time series needs to be separated to obtain approximately independent observations. The use of the advanced theory for stationary series allows to avoid the entire separation procedure. In this context the commonly applied separation techniques turn out to be quite inappropriate in most cases and the estimates based on theory of stationary series are obtained with better precision.
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Extreme Value Distributions with Applications
Fusek, Michal ; Skalská,, Hana (referee) ; Karpíšek, Zdeněk (referee) ; Michálek, Jaroslav (advisor)
The thesis is focused on extreme value distributions and their applications. Firstly, basics of the extreme value theory for one-dimensional observations are summarized. Using the limit theorem for distribution of maximum, three extreme value distributions (Gumbel, Fréchet, Weibull) are introduced and their domains of attraction are described. Two models for parametric functions estimation based on the generalized extreme value distribution (block maxima model) and the generalized Pareto distribution (threshold model) are introduced. Parameters estimates of these distributions are derived using the method of maximum likelihood and the probability weighted moment method. Described methods are used for analysis of the rainfall data in the Brno Region. Further attention is paid to Gumbel class of distributions, which is frequently used in practice. Methods for statistical inference of multiply left-censored samples from exponential and Weibull distribution considering the type I censoring are developed and subsequently used in the analysis of synthetic musk compounds concentrations. The last part of the thesis deals with the extreme value theory for two-dimensional observations. Demonstrational software for the extreme value distributions was developed as a part of this thesis.
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Statistical Analysis of Extreme Value Distributions for Censored Data
Chabičovský, Martin ; Karpíšek, Zdeněk (referee) ; Michálek, Jaroslav (advisor)
The thesis deals with extreme value distributions and censored samples. Theoretical part describes a maximum likelihood method, types of censored samples and introduce a extreme value distributions. In the thesis are derived likelihood equations for censored samples from exponential, Weibull, lognormal, Gumbel and generalized extreme value distribution. For these distributions are also derived asymptotic interval estimates and is made simulation studies on the dependence of the parameter estimate on the percentage of censoring.
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Extreme Value Distribution Parameter Estimation and its Application
Holešovský, Jan ; Picek,, Jan (referee) ; Antoch,, Jaromír (referee) ; Michálek, Jaroslav (advisor)
The thesis is focused on extreme value theory and its applications. Initially, extreme value distribution is introduced and its properties are discussed. At this basis are described two models mostly used for an extreme value analysis, i.e. the block maxima model and the Pareto-distribution threshold model. The first one takes advantage in its robustness, however recently the threshold model is mostly preferred. Although the threshold choice strongly affects estimation quality of the model, an optimal threshold selection still belongs to unsolved issues of this approach. Therefore, the thesis is focused on techniques for proper threshold identification, mainly on adaptive methods suitable for the use in practice. For this purpose a simulation study was performed and acquired knowledge was applied for analysis of precipitation records from South-Moravian region. Further on, the thesis also deals with extreme value estimation within a stationary series framework. Usually, an observed time series needs to be separated to obtain approximately independent observations. The use of the advanced theory for stationary series allows to avoid the entire separation procedure. In this context the commonly applied separation techniques turn out to be quite inappropriate in most cases and the estimates based on theory of stationary series are obtained with better precision.
|
|
|
Extreme Value Distributions with Applications
Fusek, Michal ; Skalská,, Hana (referee) ; Karpíšek, Zdeněk (referee) ; Michálek, Jaroslav (advisor)
The thesis is focused on extreme value distributions and their applications. Firstly, basics of the extreme value theory for one-dimensional observations are summarized. Using the limit theorem for distribution of maximum, three extreme value distributions (Gumbel, Fréchet, Weibull) are introduced and their domains of attraction are described. Two models for parametric functions estimation based on the generalized extreme value distribution (block maxima model) and the generalized Pareto distribution (threshold model) are introduced. Parameters estimates of these distributions are derived using the method of maximum likelihood and the probability weighted moment method. Described methods are used for analysis of the rainfall data in the Brno Region. Further attention is paid to Gumbel class of distributions, which is frequently used in practice. Methods for statistical inference of multiply left-censored samples from exponential and Weibull distribution considering the type I censoring are developed and subsequently used in the analysis of synthetic musk compounds concentrations. The last part of the thesis deals with the extreme value theory for two-dimensional observations. Demonstrational software for the extreme value distributions was developed as a part of this thesis.
|
|
|
Non-parametric estimation of parameters of extreme value distribution
Blachut, Vít ; Popela, Pavel (referee) ; Michálek, Jaroslav (advisor)
The concern of this diploma thesis is extreme value distributions. The first part formulates and proves the limit theorem for distribution of maximum. Further there are described basic properties of class of extreme value distributions. The key role of this thesis is on non-parametric estimations of extreme value index. Primarily, Hill and moment estimator are derived, for which is, based on the results of mathematical analysis, suggested an alternative choice of optimal sample fraction using a bootstrap based method. The estimators of extreme value index are compared based on simulations from proper chosen distributions, being close to distribution of given rain-fall data series. This time series is recommended a suitable estimator and suggested choice of optimal sample fraction, which belongs to the most difficult task in the area of extreme value theory.
|
|
|
Statistical Analysis of Extreme Value Distributions for Censored Data
Chabičovský, Martin ; Karpíšek, Zdeněk (referee) ; Michálek, Jaroslav (advisor)
The thesis deals with extreme value distributions and censored samples. Theoretical part describes a maximum likelihood method, types of censored samples and introduce a extreme value distributions. In the thesis are derived likelihood equations for censored samples from exponential, Weibull, lognormal, Gumbel and generalized extreme value distribution. For these distributions are also derived asymptotic interval estimates and is made simulation studies on the dependence of the parameter estimate on the percentage of censoring.
|
|
|
Statistics of extremes
Fusek, Michal ; Neubauer, Jiří (referee) ; Michálek, Jaroslav (advisor)
The thesis deals with extreme value distributions. The theoretical part is devoted to the basics of extreme value theory and to the characterization of extreme value distributions. There is the limit theorem for distributions of the maximum formulated and characteristics of the extreme value distributions deduced. There are parameter estimates for Weibull, lognormal and exponential distributions inferred using method of maximum likelihood and method of moments. There is also the theory of censored samples described. The practical part is devoted to statistical analysis of rainfall.
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