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 JFRM  Vol.9 No.3 , September 2020
Modeling Bursts and Heavy Tails in Inter-Arrival Claims in Non-Life Insurance
Abstract: Current insurance models, assuming that inter-arrival time of claims, are distributed randomly and thus well approximated by Poisson processes. Here we provide clear proof that the timing of inter-claims fits by non-Poisson patterns, marked by rapid events, separated by long periods of inactivity. The time of inter-arrival claims will be heavy tailed, most claims will be executed quickly, while a few will have very long waiting times. We will model and analysis of insurance based on claim inter-arrival time, the time interval between two successive claims and the ability to carry out such modeling was limited by a lack of ecologically relevant data collected on claims inter-arrival. We propose a structured process behavior model based on data from Egyptian fire insurance company. Our analysis shows that claim activities can be represented by non-Poisson processes and that the subsequent distribution of inter-arrival activity times follows the Pareto distribution. These results will help researchers understand daily behavioral trends and create more sophisticated predictive models of claims.
Cite this paper: Hanafy, M. (2020) Modeling Bursts and Heavy Tails in Inter-Arrival Claims in Non-Life Insurance. Journal of Financial Risk Management, 9, 314-333. doi: 10.4236/jfrm.2020.93017.
References

[1]   Amoroso, L. (1938). Vilfredo Pareto. Econometrica, 6, 1-21.
https://doi.org/10.2307/1910081

[2]   Anderson, H. R. (2003). Fixed Broadband Wireless System Design. New York: Wiley.
https://doi.org/10.1002/0470861290

[3]   Andriani, P., & McKelvey, B. (2009). Perspective—From Gaussian to Paretian Thinking: Causes and Implications of Power Laws in Organizations. Organization Science, 20, 1053-1071.
https://doi.org/10.1287/orsc.1090.0481

[4]   Arnold, B. C. (1983). Pareto Distributions. Fairland, MD: International Cooperative Publishing House.

[5]   Bees, A., York, N., & Barabasi, A. (2005). The Origin of Bursts and Heavy Tails in Human Dynamics. Nature, 435, 207-211.
https://doi.org/10.1038/nature03459

[6]   Billingsley, P. (1968). Convergence of Probability Measures. New York: Wiley.

[7]   Bingham, N. H., Goldie, C. M., & Teugels, J. L. (1987). Regular Variation. Cambridge: Cambridge University Press.
https://doi.org/10.1017/CBO9780511721434

[8]   Delampady, M., Krishnan, T., & Ramasubramanian, S. (2001). Probability and Statistics. A Volume in “Echoes from Resonance”, Hyderabad: Universities Press.

[9]   Dewes, C., Wichmann, A., & Feldman, A. (2003). Proceedings of the 2003 ACM SIGCOMM Conference on Internet Measurement (IMC-03). New York: ACM.

[10]   Ebel, H., Mielsch, L.-I., & Bornholdt, S. (2002). Scale-Free Topology of E-Mail Networks. Physical Review E, 66, R35103.

[11]   Eckmann, J.-P., Moses, E., & Sergi, D. (2004). Entropy of Dialogues Creates Coherent Structures in E-Mail Traffic. Proceedings of the National Academy of Sciences of the United States of America, 101, 14333-14337.
https://doi.org/10.1073/pnas.0405728101

[12]   Erlang, A. K. (1917) Solution of Some Problems in the Theory of Probabilities of Significance in Automatic Telephone Exchanges. Post Office Electrical Engineer’s Journal, 10, 189-197.

[13]   Ethier, S., & Kurtz, T. (1986). Markov Processes: Characterization and Convergence. New York: Wiley.
https://doi.org/10.1002/9780470316658

[14]   Feller, W. (1969). An Introduction to Probability Theory and Its Applications (Vol. II). New Delhi: Wiley-Eastern.

[15]   Feller, W. (1971). An Introduction to Probability Theory and Its Applications (Volume II, p. 704) (2 ed.). New York: John Wiley & Sons Inc.

[16]   Gallotti, R., Bazzani, A., Rambaldi, S., & Barthelemy, M. (2016). A Stochastic Model of Randomly Accelerated Walkers for Human Mobility. Nature Communications, 7, 12600.
https://doi.org/10.1038/ncomms12600

[17]   Gonzalez, M. C., Hidalgo, C. A., & Barabasi, A.-L. (2008). Understanding Individual Human Mobility Patterns. Nature, 453, 779-782.
https://doi.org/10.1038/nature06958

[18]   Grais, R. F., Ellis, J. H., & Glass, G. E. (2003). Assessing the Impact of Airline Travel on the Geographic Spread of Pandemic Influenza. European Journal of Epidemiology, 18, 1065-1072.

[19]   Haight, F. A. (1967). Handbook of the Poisson Distribution. New York: Wiley.

[20]   Henderson, S., & Henderson, E. (2001). A Note on the Public Interest and Ethical Behaviour. Australian Accounting Review, 11, 68-72.
https://doi.org/10.1111/j.1835-2561.2002.tb00391.x

[21]   Hong, S. (2010). Human Movement Patterns, Mobility Models and Their Impacts on Wireless Applications. Raleigh, NC: North Carolina State University.

[22]   Kwon, O., Son, W.-S., & Jung, W.-S. (2016). The Double Power Law in Human Collaboration Behavior: The Case of Wikipedia. Physica A: Statistical Mechanics and Its Applications, 461, 85-91.
https://doi.org/10.1016/j.physa.2016.05.010

[23]   Leskovec, J., McGlohon, M., Faloutsos, C., Glance, N., & Hurst, M. (2007). Patterns of Cascading Behavior in Large Blog Graphs. In Proceedings of the 2007 SIAM International Conference on Data Mining (pp. 551-556).
https://doi.org/10.1137/1.9781611972771.60

[24]   Li, H. et al. (2015). Characterizing Smartphone Usage Patterns from Millions of Android Users. In Proceedings of the 2015 Internet Measurement Conference (pp. 459-472).
https://doi.org/10.1145/2815675.2815686

[25]   Mainardi, F., Raberto, M., Gorenflo, R., & Scalas, E. (2000). Fractional Calculus and Continuous-Time Finance II: The Waiting-Time Distribution. Physica A: Statistical Mechanics and Its Applications, 287, 468-481.
https://doi.org/10.1016/S0378-4371(00)00386-1

[26]   Masoliver, J., Montero, M., & Weiss, G. H. (2003). Continuous-Time Random-Walk Model for Financial Distributions. Physical Review E, 67, Article ID: 021112.
https://doi.org/10.1103/PhysRevE.67.021112

[27]   Oancea, B. (2017). Income Inequality in Romania: The Exponential-Pareto Distribution. Physica A: Statistical Mechanics and Its Applications, 469, 486-498.
https://doi.org/10.1016/j.physa.2016.11.094

[28]   Oliveira, J. G., & Barabási, A.-L. (2005). Human Dynamics: Darwin and Einstein Correspondence Patterns. Nature, 437, 1251.
https://doi.org/10.1038/4371251a

[29]   Pareto, V. (1898). Cours d’economie politique. Journal of Political Economy, 6, 549-552.
https://doi.org/10.1086/250536

[30]   Pieropan, A., Ek, C. H., & Kjellström, H. (2013). Functional Object Descriptors for Human Activity Modeling. In Robotics and Automation (ICRA), 2013 IEEE International Conference on (pp. 1282-1289).
https://doi.org/10.1109/ICRA.2013.6630736

[31]   Plerou, V., Gopikrishnan, P., Amaral, A. N., Gabaix, X., & Stanley, H. E. (2000). Economic Fluctuations and Anomalous Diffusion. Physical Review E, 62, R3023.
https://doi.org/10.1103/PhysRevE.62.R3023

[32]   Rolski, T., Schmidli, H., Schmidt, V., & Teugels, J. L. (1999). Stochastic Processes for Insurance and Finance. Chichester: Wiley.
https://doi.org/10.1002/9780470317044

[33]   Saito, Y. U., Watanabe, T., & Iwamura, M. (2007). Dolarger Firms Have More Interfirm Relationships? Physica A: Statistical Mechanics and Its Applications, 383, 158-163.
https://doi.org/10.1016/j.physa.2007.04.097

[34]   Scholz, T. M. (2015). The Human Role within Organizational Change: A Complex System Perspective. In Change Management and the Human Factor (pp. 19-31). Berlin: Springer.
https://doi.org/10.1007/978-3-319-07434-4_3

[35]   Tsompanidis, I., Zahran, A. H., & Sreenan, C. J. (2014). Mobile Network Traffic: A User Behaviour Model. In 2014 7th IFIP Wireless and Mobile Networking Conference (WMNC) (pp. 1-8).
https://doi.org/10.1109/WMNC.2014.6878862

[36]   Van Montfort, M. A. J. (1986). The Generalized Pareto Distribution Applied to Rainfall Depths. Hydrological Sciences Journal, 31, 151-162.
https://doi.org/10.1080/02626668609491037

[37]   Vazquez, A. (2005). Exact Results for the Barabási Model of Human Dynamics. Physical Review Letters, 95, Article ID: 248701.
https://doi.org/10.1103/PhysRevLett.95.248701

[38]   Yu, L., Cui, P., Song, C., Zhang, T., & Yang, S. (2017). A Temporally Heterogeneous Survival Framework with Application to Social Behavior Dynamics. In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (pp. 1295-1304).
https://doi.org/10.1145/3097983.3098189

[39]   Zhu, W.-Y., Peng, W.-C., Chen, L.-J., Zheng, K., & Zhou, X. (2015). Modeling User Mobility for Location Promotion in Location-Based Social Networks. In Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (pp. 1573-1582).
https://doi.org/10.1145/2783258.2783331

 
 
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