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An application of Markov chain Monte Carlo (MCMC) to continuous-time incurred but not yet reported (IBNYR) events

Published online by Cambridge University Press:  15 July 2016

Garfield O. Brown
Affiliation:
Statistical Laboratory, Centre for Mathematical Sciences, Cambridge CB3 0WB, UK
Winston S. Buckley*
Affiliation:
Mathematical Sciences, Bentley University, Waltham, MA 02452, USA
*
*Correspondence to: Winston S. Buckley, Centre for Mathematical Sciences, Bentley University, Waltham, MA 02452, USA. Tel: +1 203-550-9765; E-mail: winb365@hotmail.com

Abstract

We develop a Bayesian model for continuous-time incurred but not yet reported (IBNYR) events under four types of secondary data, and show that unreported events, such as claims, have a Poisson distribution with a reduced arrival parameter if event arrivals are Poisson distributed. Using insurance claims as an example of an IBNYR event, we apply Markov chain Monte Carlo (MCMC) to the continuous-time IBNYR claims model of Jewell using Type I and Type IV data. We illustrate the relative stability of the MCMC method versus the Gammoid approximation of Jewell by showing that the MCMC estimates approach their prior parameters, while the Gammoid approximations grow without bound for Type IV data. Moreover, this holds for any distribution that the delay parameter is assumed to follow. Our framework also allows for the computation of posterior confidence intervals for the parameters.

Type
Papers
Copyright
© Institute and Faculty of Actuaries 2016 

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References

Gelman, A. (1996). Inference and monitoring convergence. In W.R. Gilks, S. Richardson & D. J. Spiegelhalter, (Eds.) Markov Chain Monte Mario in Practice (pp. 131144). Chapman and Hall, London.Google Scholar
Jewell, W.S. (1989). Predicting IBNYR events and delays I. Continuous time. ASTIN Bulletin, 19(1), 2555.CrossRefGoogle Scholar
Jewell, W.S. (1990). Predicting IBNYR events and delays II. Discrete time. ASTIN Bulletin, 20(1), 93111.CrossRefGoogle Scholar
Kingman, J.F.C. (1993). Poisson Processes. Oxford University Press, New York.Google Scholar