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GRIN Verlag Entscheidungen in der Bayes-Statistik und Sequentialanalyse bei unscharfer Information: Am Beispiel unscharfer Stichproben von Poisson-verteilten ... unscharfer A-posteriori-Gamma-Verteilungen

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Description

Doktorarbeit / Dissertation aus dem Jahr 2006 im Fachbereich Mathematik - Analysis, Note: 1,0, Leopold-Franzens-Universität Innsbruck (Institut für Mathematik), Sprache: Deutsch, Abstract: Die Anzahl der Flutwellen kritischer Höhe in einem Zeitintervall kann als Poisson-verteilte Zufallsgröße aufgefasst, da diskrete, zufällig über die Zeit verteilte, mit konstanter Häufigkeit auftretende Ereignisse gezählt werden. Die Höhe einer Flutwelle ist aber eine unscharfe Größe. Geeignete Instrumente zur Erweiterung von mathematischen Analysemethoden von unscharfen Sachverhalten stellt die Theorie der Fuzzy-Mengen zur Verfügung. Als von unscharfen Werten abgeleitete Größe ist auch die Anzahl der Flutwellen kritischer Höhe in einer Beobachtungsperiode unscharf, insbesondere stellt sie eine unscharfe Teilmenge der nichtnegativen ganzen Zahlen dar und kann als Realisierung einer unscharf Poisson-verteilten Fuzzy-Zufallsvariablen angesehen werden. Der unscharfe Zählprozess ist somit ein unscharfer Poisson-Prozess. Die Verfahren der klassischen Inferenzstatistik lassen sich mit Hilfe des Extensionsprinzips auf unscharfe Realisationen von Stichproben von Fuzzy-Zufallsvariablen erweitern. Bei Vorliegen einer geeigneten zur Stichprobenverteilung konjugierten Verteilungsfamilie ist eine einfache Erweiterung des Bayes'schen Theorems auf unscharfe Information möglich, so kann eine exakte oder unscharfe A-priori-Gamma-Verteilung und eine unscharfe Stichproben-Poisson-Verteilung zu einer unscharfen A-posteriori-Gamma-Verteilung kombiniert werden. Ausgehend von der unscharfen A-posteriori-Verteilung kann gezeigt werden, dass Anwendung des Extensionsprinzips auf Bayes'schen verlustminimierenden Entscheidungsregeln zu unscharfen Entscheidungen führt, die im Sinne einer geeigneten Optimalitätsdefinition als optimal angesehen werden können. Bei sequentiellen statistischen Entscheidungsverfahren wird der für die statistische Entscheidung benötigte Stichprobenumfang (Stoppzeit) während des Beoba

Product Specifications

Format
paperback
Domain
Amazon UK
Release Date
09 October 2014
Listed Since
10 October 2014

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