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On the modelling of multivariate counts with Cox processes and dependent shot noise intensities

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  • Benjamin Avanzi
  • Gregory Clive Taylor
  • Bernard Wong
  • Xinda Yang

Abstract

In this paper, we develop a method to model and estimate several, _dependent_ count processes, using granular data. Specifically, we develop a multivariate Cox process with shot noise intensities to jointly model the arrival process of counts (e.g. insurance claims). The dependency structure is introduced via multivariate shot noise _intensity_ processes which are connected with the help of L\'evy copulas. In aggregate, our approach allows for (i) over-dispersion and auto-correlation within each line of business; (ii) realistic features involving time-varying, known covariates; and (iii) parsimonious dependence between processes without requiring simultaneous primary (e.g. accidents) events. The explicit incorporation of time-varying, known covariates can accommodate characteristics of real data and hence facilitate implementation in practice. In an insurance context, these could be changes in policy volumes over time, as well as seasonality patterns and trends, which may explain some of the relationship (dependence) between multiple claims processes, or at least help tease out those relationships. Finally, we develop a filtering algorithm based on the reversible-jump Markov Chain Monte Carlo (RJMCMC) method to estimate the latent stochastic intensities and illustrate model calibration using real data from the AUSI data set.

Suggested Citation

  • Benjamin Avanzi & Gregory Clive Taylor & Bernard Wong & Xinda Yang, 2020. "On the modelling of multivariate counts with Cox processes and dependent shot noise intensities," Papers 2004.11169, arXiv.org, revised Dec 2020.
    • Handle: RePEc:arx:papers:2004.11169
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    References listed on IDEAS

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    1. Zhao, Xiao Bing & Zhou, Xian & Wang, Jing Long, 2009. "Semiparametric model for prediction of individual claim loss reserving," Insurance: Mathematics and Economics, Elsevier, vol. 45(1), pages 1-8, August.
    2. Alexander J. McNeil & Rüdiger Frey & Paul Embrechts, 2015. "Quantitative Risk Management: Concepts, Techniques and Tools Revised edition," Economics Books, Princeton University Press, edition 2, number 10496.
    3. Larsen, Christian Roholte, 2007. "An Individual Claims Reserving Model," ASTIN Bulletin, Cambridge University Press, vol. 37(1), pages 113-132, May.
    4. Pigeon, Mathieu & Antonio, Katrien & Denuit, Michel, 2014. "Individual loss reserving using paid–incurred data," LIDAM Reprints ISBA 2014024, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    5. Pigeon, Mathieu & Antonio, Katrien & Denuit, Michel, 2013. "Individual Loss Reserving With The Multivariate Skew Normal Framework," ASTIN Bulletin, Cambridge University Press, vol. 43(3), pages 399-428, September.
    6. Zhao, XiaoBing & Zhou, Xian, 2010. "Applying copula models to individual claim loss reserving methods," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 290-299, April.
    7. Benjamin Avanzi & Greg Taylor & Bernard Wong & Alan Xian, 2020. "Modelling and understanding count processes through a Markov-modulated non-homogeneous Poisson process framework," Papers 2003.13888, arXiv.org, revised May 2020.
    8. Avanzi, Benjamin & Cassar, Luke C. & Wong, Bernard, 2011. "Modelling Dependence in Insurance Claims Processes with Lévy Copulas," ASTIN Bulletin, Cambridge University Press, vol. 41(2), pages 575-609, November.
    9. Badescu, Andrei L. & Lin, X. Sheldon & Tang, Dameng, 2016. "A marked Cox model for the number of IBNR claims: Theory," Insurance: Mathematics and Economics, Elsevier, vol. 69(C), pages 29-37.
    10. Huang, Jinlong & Wu, Xianyi & Zhou, Xian, 2016. "Asymptotic behaviors of stochastic reserving: Aggregate versus individual models," European Journal of Operational Research, Elsevier, vol. 249(2), pages 657-666.
    11. Taylor, Greg & McGuire, Gráinne & Sullivan, James, 2008. "Individual Claim Loss Reserving Conditioned by Case Estimates," Annals of Actuarial Science, Cambridge University Press, vol. 3(1-2), pages 215-256, September.
    12. Shi, Peng & Frees, Edward W., 2011. "Dependent Loss Reserving using Copulas," ASTIN Bulletin, Cambridge University Press, vol. 41(2), pages 449-486, November.
    13. Lindskog, Filip & McNeil, Alexander J., 2003. "Common Poisson Shock Models: Applications to Insurance and Credit Risk Modelling," ASTIN Bulletin, Cambridge University Press, vol. 33(2), pages 209-238, November.
    14. Pigeon, Mathieu & Antonio, Katrien & Denuit, Michel, 2014. "Individual loss reserving using paid–incurred data," Insurance: Mathematics and Economics, Elsevier, vol. 58(C), pages 121-131.
    15. Jinlong Huang & Chunjuan Qiu & Xianyi Wu, 2015. "Stochastic Loss Reserving in Discrete Time: Individual vs. Aggregate Data Models," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 44(10), pages 2180-2206, May.
    16. Merz, M. & Wüthrich, M. V., 2007. "Prediction Error of the Chain Ladder Reserving Method applied to Correlated Run-off Triangles," Annals of Actuarial Science, Cambridge University Press, vol. 2(1), pages 25-50, March.
    17. Merz, Michael & Wüthrich, Mario V. & Hashorva, Enkelejd, 2013. "Dependence modelling in multivariate claims run-off triangles," Annals of Actuarial Science, Cambridge University Press, vol. 7(1), pages 3-25, March.
    18. Matsui, Muneya, 2014. "Prediction in a non-homogeneous Poisson cluster model," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 10-17.
    19. Norberg, Ragnar, 1993. "Prediction of Outstanding Liabilities in Non-Life Insurance1," ASTIN Bulletin, Cambridge University Press, vol. 23(1), pages 95-115, May.
    20. Avanzi, Benjamin & Taylor, Greg & Wong, Bernard, 2016. "Correlations Between Insurance Lines Of Business: An Illusion Or A Real Phenomenon? Some Methodological Considerations," ASTIN Bulletin, Cambridge University Press, vol. 46(2), pages 225-263, May.
    21. Avanzi, Benjamin & Wong, Bernard & Yang, Xinda, 2016. "A micro-level claim count model with overdispersion and reporting delays," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 1-14.
    22. Badescu, Andrei L. & Chen, Tianle & Lin, X. Sheldon & Tang, Dameng, 2019. "A Marked Cox Model For The Number Of Ibnr Claims: Estimation And Application," ASTIN Bulletin, Cambridge University Press, vol. 49(3), pages 709-739, September.
    23. Centanni, Silvia & Minozzo, Marco, 2006. "A Monte Carlo Approach to Filtering for a Class of Marked Doubly Stochastic Poisson Processes," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1582-1597, December.
    24. Norberg, Ragnar, 1999. "Prediction of Outstanding Liabilities II. Model Variations and Extensions," ASTIN Bulletin, Cambridge University Press, vol. 29(1), pages 5-25, May.
    25. Arjas, Elja, 1989. "The Claims Reserving Problem in Non-Life Insurance: Some Structural Ideas," ASTIN Bulletin, Cambridge University Press, vol. 19(2), pages 139-152, November.
    26. Michael Merz & Mario Wüthrich, 2008. "Prediction Error of the Multivariate Chain Ladder Reserving Method," North American Actuarial Journal, Taylor & Francis Journals, vol. 12(2), pages 175-197.
    27. Avanzi, Benjamin & Tao, Jamie & Wong, Bernard & Yang, Xinda, 2016. "Capturing non-exchangeable dependence in multivariate loss processes with nested Archimedean Lévy copulas," Annals of Actuarial Science, Cambridge University Press, vol. 10(1), pages 87-117, March.
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