A Bayesian Inference of Non-Life Insurance Based on Claim Counting Process with Periodic Claim Intensity
ABSTRACT
The aim of this study is to propose an estimation approach to non-life insurance claim counts related to the insurance claim counting process, including the non-homogeneous Poisson process (NHPP) with a bell-shaped intensity and a beta-shaped intensity. The estimating function, such as the zero mean martingale (ZMM), is used as a procedure for parameter estimation of the insurance claim counting process, and the parameters of model claim intensity are estimated by the Bayesian method. Then,Λ(t), the compensator of N(t) is proposed for the number of claims in a time interval (0,t]. Given the process over the time interval (0,t]., the situations are presented through a simulation study and some examples of these situations are also depicted by a sample path relating N(t) to its compensatorΛ(t).
The aim of this study is to propose an estimation approach to non-life insurance claim counts related to the insurance claim counting process, including the non-homogeneous Poisson process (NHPP) with a bell-shaped intensity and a beta-shaped intensity. The estimating function, such as the zero mean martingale (ZMM), is used as a procedure for parameter estimation of the insurance claim counting process, and the parameters of model claim intensity are estimated by the Bayesian method. Then,Λ(t), the compensator of N(t) is proposed for the number of claims in a time interval (0,t]. Given the process over the time interval (0,t]., the situations are presented through a simulation study and some examples of these situations are also depicted by a sample path relating N(t) to its compensatorΛ(t).
Cite this paper
U. Jaroengeratikun, W. Bodhisuwan and A. Thongteeraparp, "A Bayesian Inference of Non-Life Insurance Based on Claim Counting Process with Periodic Claim Intensity," Open Journal of Statistics, Vol. 2 No. 2, 2012, pp. 177-183. doi: 10.4236/ojs.2012.22020.
U. Jaroengeratikun, W. Bodhisuwan and A. Thongteeraparp, "A Bayesian Inference of Non-Life Insurance Based on Claim Counting Process with Periodic Claim Intensity," Open Journal of Statistics, Vol. 2 No. 2, 2012, pp. 177-183. doi: 10.4236/ojs.2012.22020.
References
[1] W. S. Jewell, “Credible Means Are Exact Bayesian for Exponential Families,” ASTIN Bulletin, Vol. 8, No. 1, 1974, pp. 77-90. [2] R. Kaas, D. Dannenburg and M. Goovaerts, “Exact Credibility for Weighted Observations,” ASTIN Bulletin, Vol. 27, No. 2, 1997, pp. 287-295. doi:10.2143/AST.27.2.542053 [3] E. Ohlsson and B. Johansson, “Exact Credibility and Tweedie Models,” ASTIN Bulletin, Vol. 36, No. 1, 2006, pp. 121-133. doi:10.2143/AST.36.1.2014146 [4] T. Mikosch, “Non-Life Insurance Mathematics,” 2nd Edition, Springer-Verlag, Berlin, 2009. doi:10.1007/978-3-540-88233-6 [5] 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. [6] Y. Lu and J. Garrido, “Doubly Periodic Non-Homogeneous Poisson Models for Hurricane Data,” Statistical Methodology, Vol. 2, No. 1, 2005, pp. 17-35. doi:10.1016/j.stamet.2004.10.004 [7] J. Garrido and Y. Lu, “On Double Pariodic Non-Homo- geneous Poisson Processes,” Bulletin of the Association of Swiss Actuaries, Vol. 2, 2004, pp. 195-212. [8] 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 [9] P. Mukhopadhyay, “An Introduction to Estimating Functions,” Alpha Science International Ltd., Harrow, 2004. [10] P. K. Andersen, O. Borgan, R. D. Gill and N. Keiding, “Statistical Models Based on Counting Processes,” Springer-Verlag New York, Inc., New York, 1993. [11] 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 [12] 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 [13] J. Gill, “Bayesian Methods: A Social and Behavioral Sciences Approach,” Chapman and Hall/CRC, London, 2008. [14] T. Hirata, H. Okamura and T. Dohi, “A Bayesian Inference Tool for NHPP-Based Software Reliability Assessment,” In Y. H. Lee, T.-H. Kim, W.-C. Fang and D. Slezak, Eds., Lecture Notes in Computer Science, Vol. 5899, Springer, Berlin, 2009, pp. 225-236. [15] D. P. M. Scollnik, “Actuarial Modeling with MCMC and BUGS,” North American Actuarial Journal, Vol. 5, No. 2, 2001, pp. 96-124. [16] X. Zhao, C. Yu and H. Tong, “A Bayesian Approach to Weibull Survival Model for Clinical Randomized Censoring Trial based on MCMC Simulation,” The 2nd International Conference on Bioinformatics and Biomedical Engineering, Shanghai, 16-18 May 2008, pp. 1181-1184.
[1] W. S. Jewell, “Credible Means Are Exact Bayesian for Exponential Families,” ASTIN Bulletin, Vol. 8, No. 1, 1974, pp. 77-90. [2] R. Kaas, D. Dannenburg and M. Goovaerts, “Exact Credibility for Weighted Observations,” ASTIN Bulletin, Vol. 27, No. 2, 1997, pp. 287-295. doi:10.2143/AST.27.2.542053 [3] E. Ohlsson and B. Johansson, “Exact Credibility and Tweedie Models,” ASTIN Bulletin, Vol. 36, No. 1, 2006, pp. 121-133. doi:10.2143/AST.36.1.2014146 [4] T. Mikosch, “Non-Life Insurance Mathematics,” 2nd Edition, Springer-Verlag, Berlin, 2009. doi:10.1007/978-3-540-88233-6 [5] 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. [6] Y. Lu and J. Garrido, “Doubly Periodic Non-Homogeneous Poisson Models for Hurricane Data,” Statistical Methodology, Vol. 2, No. 1, 2005, pp. 17-35. doi:10.1016/j.stamet.2004.10.004 [7] J. Garrido and Y. Lu, “On Double Pariodic Non-Homo- geneous Poisson Processes,” Bulletin of the Association of Swiss Actuaries, Vol. 2, 2004, pp. 195-212. [8] 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 [9] P. Mukhopadhyay, “An Introduction to Estimating Functions,” Alpha Science International Ltd., Harrow, 2004. [10] P. K. Andersen, O. Borgan, R. D. Gill and N. Keiding, “Statistical Models Based on Counting Processes,” Springer-Verlag New York, Inc., New York, 1993. [11] 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 [12] 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 [13] J. Gill, “Bayesian Methods: A Social and Behavioral Sciences Approach,” Chapman and Hall/CRC, London, 2008. [14] T. Hirata, H. Okamura and T. Dohi, “A Bayesian Inference Tool for NHPP-Based Software Reliability Assessment,” In Y. H. Lee, T.-H. Kim, W.-C. Fang and D. Slezak, Eds., Lecture Notes in Computer Science, Vol. 5899, Springer, Berlin, 2009, pp. 225-236. [15] D. P. M. Scollnik, “Actuarial Modeling with MCMC and BUGS,” North American Actuarial Journal, Vol. 5, No. 2, 2001, pp. 96-124. [16] X. Zhao, C. Yu and H. Tong, “A Bayesian Approach to Weibull Survival Model for Clinical Randomized Censoring Trial based on MCMC Simulation,” The 2nd International Conference on Bioinformatics and Biomedical Engineering, Shanghai, 16-18 May 2008, pp. 1181-1184.