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Confidence Intervals for Quantiles
Horejšová, Markéta ; Kulich, Michal (advisor) ; Hlávka, Zdeněk (referee)
In this thesis, various construction methods for simultaneous confidence intervals for quantiles are explained. Among nonparametric approaches, a special emphasis is dedicated to a recent method based on a multinomial distribution for calculating the overall confidence level of confidence intervals for all quantiles of interest using an efficient recursive algorithm, which is also described. Furthermore, a method based on Kolmogorov-Smirnov statistic or an asymptotic method using empirical distribution function and order statistics for quantile estimate are presented. A special parametric method for several quantiles of a normally distributed population is introduced along with a few of its modifications. Subsequently, a simulation is run to test the real coverage of the described theoretical methods. Powered by TCPDF (www.tcpdf.org)
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Multistage nested distance in stochastic optimization
Horejšová, Markéta ; Vitali, Sebastiano (advisor) ; Lachout, Petr (referee)
Multistage stochastic optimization is used to solve many real-life problems where decisions are taken at multiple times, e.g., portfolio selection problems. Such problems need the definition of stochastic processes, which are usually approxim- ated by scenario trees. The choice of the size of the scenario trees is the result of a compromise between the best approximation and the possibilities of the com- puter technology. Therefore, once a master scenario tree has been generated, it can be needed to reduce its dimension in order to make the problem computation- ally tractable. In this thesis, we introduce several scenario reduction algorithms and we compare them numerically for different types of master trees. A simple portfolio selection problem is also solved within the study. The distance from the initial scenario tree, the computational time, and the distance between the optimal objective values and solutions are compared for all the scenario reduction algorithms. In particular, we adopt the nested distance to measure the distance between two scenario trees. 1
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Confidence Intervals for Quantiles
Horejšová, Markéta ; Kulich, Michal (advisor) ; Hlávka, Zdeněk (referee)
In this thesis, various construction methods for simultaneous confidence intervals for quantiles are explained. Among nonparametric approaches, a special emphasis is dedicated to a recent method based on a multinomial distribution for calculating the overall confidence level of confidence intervals for all quantiles of interest using an efficient recursive algorithm, which is also described. Furthermore, a method based on Kolmogorov-Smirnov statistic or an asymptotic method using empirical distribution function and order statistics for quantile estimate are presented. A special parametric method for several quantiles of a normally distributed population is introduced along with a few of its modifications. Subsequently, a simulation is run to test the real coverage of the described theoretical methods. Powered by TCPDF (www.tcpdf.org)
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