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Hable, Robert (2009): Data-Based Decisions under Complex Uncertainty. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics
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Abstract

Decision theory is, in particular in economics, medical expert systems and statistics, an important tool for determining optimal decisions under uncertainty. In view of applications in statistics, the present book is concerned with decision problems which are explicitly data-based. Since the arising uncertainties are often too complex to be described by classical precise probability assessments, concepts of imprecise probabilities (coherent lower previsions, F-probabilities) are applied. Due to the present state of research, some basic groundwork has to be done: Firstly, topological properties of different concepts of imprecise probabilities are investigated. In particular, the concept of coherent lower previsions appears to have advantageous properties for applications in decision theory. Secondly, several decision theoretic tools are developed for imprecise probabilities. These tools are mainly based on concepts developed by L. Le Cam and enable, for example, a definition of sufficiency in case of imprecise probabilities for the first time. Building on that, the article [A. Buja, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 65 (1984) 367-384] is reinvestigated in the only recently available framework of imprecise probabilities. This leads to a generalization of results within the Huber-Strassen theory concerning least favorable pairs or models. Results obtained by these investigations can also be applied afterwards in order to justify the use of the method of natural extension, which is fundamental within the theory of imprecise probabilities, in data-based decision problems. It is shown by means of the theory of vector lattices that applying the method of natural extension in decision problems does not affect the optimality of decisions. However, it is also shown that, in general, the method of natural extension suffers from a severe instability. The book closes with an application in statistics in which a minimum distance estimator is developed for imprecise probabilities. After an investigation concerning its asymptotic properties, an algorithm for calculating the estimator is given which is based on linear programming. This algorithm has led to an implementation of the estimator in the programming language R which is publicly available as R package "imprProbEst". The applicability of the estimator (even for large sample sizes) is demonstrated in a simulation study.