A Statistical Analysis of Intensities Estimation on the Modeling of Non-Life Insurance Claim Counting Process

Uraiwan Jaroengeratikun, Winai Bodhisuwan, Ampai Thongteeraparp

.

DOI: 10.4236/am.2012.31016                

Abstract

This study presents an estimation approach to non-life insurance claim counts relating to a specified time. The objective of this study is to estimate the parameters in non-life insurance claim counting process, including the homogeneous Poisson process (HPP) and the non-homogeneous Poisson process (NHPP) with a bell-shaped intensity. We use the estimating function, the zero mean martingale (ZMM) as a procedure of parameter estimation in the insurance claim counting process. Then, Λ(t) , the compensator of is proposed for the number of claims in the time interval . We present situations through a simulation study of both processes on the time interval . Some examples of the situations in the simulation study are depicted by a sample path relating to its compensator Λ(t). In addition, an example of the claim counting process illustrates the result of the compensator estimate misspecification.

Keywords

Estimating Function; Zero Mean Martingale; Non-Life Insurance Claim Counting Process; Poisson Process; Bell-Shaped Intensity

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	S. A. Klugman, H. H. Panjer and G. E. Willmot, “Loss Models from Data to Decisions,” 3rd Edition, John Wiley & Sons, Hoboken, 2008.
[2]	M. Denuit, X. Maréchal, S. Pitrebois and J. F. Walhin, “Actuarial Modelling of Claim Counts,” John Wiley & Sons, Hoboken, 2007. doi:10.1002/9780470517420
[3]	H. Bühlmann, “Introduction Report Experience Rating and Credibility,” ASTIN Bulletin, Vol. 4, No. 3, 1967, pp. 199-207.
[4]	H. Bühlmann, “Credibility Procedures,” Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, Vol. 1, 1972, pp. 515-525.
[5]	T. Mikosch, “Non-Life Insurance Mathematics,” 2nd Edi- tion, Springer-Verlag, Berlin, 2009. doi:10.1007/978-3-540-88233-6
[6]	M. Matsui and T. Mikosch, “Prediction in a Poisson Cluster Model,” Journal of Applied Probability, Vol. 47, No. 2, 2010, pp. 350-366. doi:10.1239/jap/1276784896
[7]	M. Morales, “On a Surplus Process under a Periodic Environment: A Simulation Approach,” North American Actuarial Journal, Vol. 8, No. 2, 2004, pp. 76-87.
[8]	Y. Lu and J. Garrido, “On Double Periodic Non-Homogeneous Poisson Processes,” Bulletin of the Swiss Association of Actuaries Swiss Association of Actuaries-Bern, Bern, 2004, pp. 195-212.
[9]	S. M. Ross, “Introduction to Probability Models,” 5th Edition, Academic Press, Inc., San Diego, 1993.
[10]	P. Mukhopadhyay, “An Introduction to Estimating Functions,” Alpha Science International Ltd., Harrow, 2004.
[11]	P. K. Andersen, O. Borgan, R. D. Gill and N. Keiding, “Statistical Models Based on Counting Processes,” Springer-Verlag, New York, 1993.
[12]	P. Yip, “Estimating the Number of Error in a System Using a Martingale Approach,” IEEE Transactions on Reliability, Vol. 44, No. 2, 1995. pp. 322-326. doi:10.1109/24.387389
[13]	J. E. R. Cid and J. A. Achcar, “Bayesian Inference for Nonhomogeneous Poisson Processed in Software Reliability Models Assuming Nonmonotonic Intensity Functions,” Computational Statistics & Data Analysis, Vol. 32, No. 2, 1999, pp.147-159. doi:10.1016/S0167-9473(99)00028-6