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

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.

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.

Cite this paper

U. Jaroengeratikun, W. Bodhisuwan and A. Thongteeraparp, "A Statistical Analysis of Intensities Estimation on the Modeling of Non-Life Insurance Claim Counting Process,"*Applied Mathematics*, Vol. 3 No. 1, 2012, pp. 100-106. doi: 10.4236/am.2012.31016.

U. Jaroengeratikun, W. Bodhisuwan and A. Thongteeraparp, "A Statistical Analysis of Intensities Estimation on the Modeling of Non-Life Insurance Claim Counting Process,"

References

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[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

[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