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Confidence regions in nonlinear regression
Marcinko, Tomáš ; Zvára, Karel (advisor) ; Komárek, Arnošt (referee)
The aim of this thesis is a comprehensive description of the properties of a nonlinear least squares estimator for a nonlinear regression model with normally distributed errors and thorough development of various methods for constructing confidence regions and confidence intervals for the parameters of the nonlinear model. Due to the fact that, unlike the case of linear models, there is no easy way to construct an exact confidence region for the parameters, most of these methods are only approximate. A short simulation study comparing observed coverage of various confidence regions and confidence intervals for models with different curvatures and sample sizes is also included. In case of negligible intrinsic curvature the use of likelihood-ratio confidence regions seems the most appropriate.
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Confidence regions in nonlinear regression
Marcinko, Tomáš ; Zvára, Karel (advisor) ; Komárek, Arnošt (referee)
The aim of this thesis is a comprehensive description of the properties of a nonlinear least squares estimator for a nonlinear regression model with normally distributed errors and thorough development of various methods for constructing confidence regions and confidence intervals for the parameters of the nonlinear model. Due to the fact that, unlike the case of linear models, there is no easy way to construct an exact confidence region for the parameters, most of these methods are only approximate. A short simulation study comparing observed coverage of various confidence regions and confidence intervals for models with different curvatures and sample sizes is also included. In case of negligible intrinsic curvature the use of likelihood-ratio confidence regions seems the most appropriate.
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Consequences of assumption violations of selected statistical methods
Marcinko, Tomáš ; Blatná, Dagmar (advisor) ; Malá, Ivana (referee) ; Lukáš, Ladislav (referee)
Classical parametric methods of statistical inference and hypothesis testing are derived under fundamental theoretical assumptions, which may or may not be met in real world applications. However, these methods are usually used despite the violation of their underlying assumptions, while it is argued, that these methods are quite insensitive to the violation of relevant assumptions. Moreover, alternative nonparametric or rank tests are often overlooked, mostly because these methods may be deemed to be less powerful then parametric methods. The aim of the dissertation is therefore a description of the consequences of assumption violations concerning classical one-sample and two-sample statistical methods and a consistent and comprehensive comparison of parametric, nonparametric and robust statistical techniques, which is based on extensive simulation study and focused mostly on a normality and heteroscedasticity assumption violation. The results of the simulation study confirmed that the classical parametric methods are relatively robust, with some reservations in case of outlying observations, when traditional methods may fail. On the other hand, the empirical study clearly proved that the classical parametric methods are losing their optimal properties, when the underlying assumptions are violated. For example, in many cases of non-normality the appropriate nonparametric and rank-based methods are more powerful, and therefore a statement, that these methods are unproductive due to their lack of power may be considered a crucial mistake. However, the choice of the most appropriate distribution-free method generally depends on the particular form of the underlying distribution.
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Split delivery vehicle routing problem
Marcinko, Tomáš ; Pelikán, Jan (advisor) ; Fábry, Jan (referee)
This thesis focuses on a description of the split delivery vehicle routing problem (SDVRP), in which the restriction that each customer has to be visited exactly once is not assumed, contrary to the classical vehicle routing problem, and split deliveries are allowed. Considering the fact that the split delivery vehicle routing problem in NP-hard, a number of heuristic algorithms proposed in the literature are presented. Computational experiments are reported and the results show that the largest benefits of split deliveries are obtained in case of instances with fairly specific characteristics and also several drawbacks of implemented Tabu Search algorithm (SPLITABU) are point out.
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