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Einführung zu Markov Chain Monte Carlo Verfahren mit Anwendung auf Gesamtschadenmodelle

Introduction to Markov Chain Monte Carlo methods with application to aggregate insurance loss

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Blätter der DGVFM

Zusammenfassung

Modellierung von realistischen Zusammenhängen erfordern komplexe statistische Modelle. Standard Schätzmethoden wie Maximum Likelihood sind dann oft auch numerisch nicht mehr verwendbar, während eine Bayesianische Schätzung noch durchführbar ist. Dies ist möglich durch das Einsetzen von Markov Chain Monte Carlo (MCMC) Methoden. Dieser Artikel beschreibt in Grundzügen wie MCMC Verfahren arbeiten und diskutiert zwei Standardverfahren, nämlich den Gibbs Sampler und den Metropolis Hastings Algorithmus. Praktische Hinweise zur Implementierung und Auswertung werden gegeben. Dabei wird die Software (win)bugs eingesetzt. Zum Abschluss wird die Implementierung von MCMC Algorithmen und deren Bayesianische Auswertung an einem einfachen statistischen Modells illustriert. Dabei wurden Standardmodelle wie das Pareto-Poisson und das GPD-Poisson Modell für die Modellierung des Gesamtschaden in der Versicherungswirtschaft eingesetzt und die Parameterunsicherheit bei der Vorhersage berücksichtigt.

Summary

Modeling of realistic dependency structures requires the usage of complex statistical models. In this case standard estimating methods such as maximum likelihood are not even numerically tractable in contrast to Bayesian approaches. This is facilitated by utilising Markov Chain Monte Carlo (MCMC) methods. This paper introduces MCMC approaches such as the Gibbs sampler and the Metropolis Hastings algorithm. Practical advice with regard to implementation and evaluation will be given. The statistical software (win)bugs will be used. Finally a data example requiring modeling of aggregate insurance loss will be discussed in detail when parameter uncertainty is accounted for. In particular standard models such as the Pareto-Poisson and the GPD-Poisson models were used to illustrate the Bayesian approach with MCMC methods.

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Authors and Affiliations

  1. München

    Claudia Czado

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  1. Claudia Czado
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basiert auf einem eingeladenen Vortrag gehalten anlässlich derScientific Conference on Insurance and Finance der Deutschen Gesellschaft für Versicherungs- und Finanzmathematik am 28. April 2003 in Bonn

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Czado, C. Einführung zu Markov Chain Monte Carlo Verfahren mit Anwendung auf Gesamtschadenmodelle. Blätter DGVFM 26, 331–350 (2004). https://doi.org/10.1007/BF02858079

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