Skip to main content
SpringerLink
Log in

A Fuzzy Least-Squares Estimation of a Hybrid Log-Poisson Regression and Its Goodness of Fit for Optimal Loss Reserves in Insurance

  • Published:
International Journal of Fuzzy Systems Aims and scope Submit manuscript

Abstract

In loss reserving, the log-Poisson regression model is a well-known stochastic model underlying the chain-ladder method which is the most used method for reserving purposes. Mack (ASTIN Bull 21(01):93–109, 1991) proved that the log-Poisson model provides the same estimates as the chain-ladder method. So in this article, our objective is to improve the log-Poisson regression model in loss reserving framework. Thereby, we prove the reliability of hybrid models in loss reserving, especially when the data contain fuzziness, for example when the claims are related to body injures (Straub and Swiss in Non-life insurance mathematics, Springer, Berlin, 1988). Thus, we estimate a hybrid generalized linear model (GLM) (log-Poisson) using the fuzzy least-squares procedures (Celmiš in Fuzzy Sets Syst 22(3):245–269, 1987a; Math Model 9(9):669–690, 1987b; D’Urso and Gastaldi in Comput Stat Data Anal 34(4): 427–440, 2000; in: Advances in classification and data analysis, Springer, 2001). We develop a new goodness of fit index to compare this new model and the classical log-Poisson regression (Mack 1991). Both the classical log-Poisson model and the hybrid one are performed on a loss reserving data. According to the goodness of fit index and the mean square error prediction, we prove that the new model provide better results than the classical log-Poisson model. This comparison can be extend to any other GLM in loss reserving.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Accomando, F. W., Weissner, E.: Report lag distributions: estimation and application to IBNR counts. In: Transcripts of the 1988 Casualty Loss Reserve Seminar, pp. 1038–1133. Casualty Actuarial Society, Arlington (1988)

  2. Adams, K. H.: Chain Ladder Reserving Methods for Liabilities with Per Occurrence Limits. In: Casualty Actuarial Society E-Forum, p. 112., Fall (2013). http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.406.7355&rep=rep1&type=pdf

  3. Asai, H.T.-S.U.-K.: Linear regression analysis with fuzzy model. IEEE Trans. Syst. Man Cybern. 12, 903–907 (1982)

    Article  MATH  Google Scholar 

  4. Barnett, G., Zehnwirth, B.: Best estimates for reserves. In: Proceedings of the Casualty Actuarial Society, vol. 87, No. 167, pp. 245–321 (2000)

  5. Björkwall, S., Hssjer, O., Ohlsson, E.: Bootstrapping the separation method in claims reserving. ASTIN Bull. J. IAA 40(2), 845–869 (2010)

    MathSciNet  MATH  Google Scholar 

  6. Blum, K.A., Otto, D.J.: Best estimate loss reserving: an actuarial perspective. In: CAS Forum Fall, vol. 1, No. 55, pp. 101 (1998)

  7. Bornhuetter, R.L., Ferguson, R.E.: The actuary and IBNR. In: Proceedings of the Casualty Actuarial Society, vol. 59, pp. 181–195 (1972)

  8. Buckley, J.J.: Fuzzy Probability and Statistics, vol. 196. Springer, Berlin (2006)

    MATH  Google Scholar 

  9. Buchwalder, M., Bhlmann, H., Merz, M., Wthrich, M.V.: The mean square error of prediction in the chain ladder reserving method (mack and murphy revisited). ASTIN Bull. J. IAA 36(2), 521–542 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Celmiš, A.: Least squares model fitting to fuzzy vector data. Fuzzy Sets Syst. 22(3), 245–269 (1987a)

    Article  MathSciNet  Google Scholar 

  11. Celmiš, A.: Multidimensional least-squares fitting of fuzzy models. Math. Model. 9(9), 669–690 (1987b)

    Article  Google Scholar 

  12. Christofides, S.: Regression models based on log-incremental payments. Claims Reserving Man v. 2, pp. D5.1-D5.53. Institute of Actuaries, London (1990)

  13. Dahms, R.: Linear stochastic reserving methods. ASTIN Bull. J. IAA 42(1), 1–34 (2012)

    MathSciNet  MATH  Google Scholar 

  14. de Andrés Sánchez, J.: Calculating insurance claim reserves with fuzzy regression. Fuzzy Sets Syst. 157(23), 3091–3108 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. de Andrés-Sánchez, J.: Claim reserving with fuzzy regression and taylors geometric separation method. Insur. Math. Econ. 40(1), 145–163 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. de Andrés-Sánchez, J.: Claim reserving with fuzzy regression and the two ways of anova. Appl. Soft Comput. 12(8), 2435–2441 (2012)

    Article  Google Scholar 

  17. de Andrés Sánchez, J.: Fuzzy claim reserving in non-life insurance. Comput. Sci. Inf. Syst. 11(2), 825–838 (2014)

    Article  Google Scholar 

  18. de Campos Ibáñez, L.M., Muñoz, A.G.: A subjective approach for ranking fuzzy numbers. Fuzzy Sets Syst. 29(2), 145–153 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dubois, D., Prade, H.: Operations on fuzzy numbers. Int. J. Syst. Sci. 9(6), 613–626 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  20. Dubois, D., Prade, H.: Fuzzy numbers: an overview. In: Readings in Fuzzy Sets for Intelligent Systems, pp. 112–148. Morgan Kaufmann Publishers (1993)

  21. D’Urso, P.: Linear regression analysis for fuzzy/crisp input and fuzzy/crisp output data. Comput. Stat. Data Anal. 42(1–2), 47–72 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. D’Urso, P., Gastaldi, T.: A least-squares approach to fuzzy linear regression analysis. Comput. Stat. Data Anal. 34(4), 427–440 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. D'Urso, P., Gastaldi, T.: Linear fuzzy regression analysis with asymmetric spreads. In : Advances in Classification and Data Analysis, pp. 257-264. Springer, Berlin, Heidelberg (2001)

  24. England, P., Verrall, R.: Analytic and bootstrap estimates of prediction errors in claims reserving. Insur. Math. Econ. 25(3), 281–293 (1999)

    Article  MATH  Google Scholar 

  25. England, P.D., Verrall, R.J.: Stochastic claims reserving in general insurance. Br. Actuar. J. 8(03), 443–518 (2002)

    Article  Google Scholar 

  26. Francis, B., Green, M., Payne, C.: The GLIM System: Release 4 Manual. Clarendon Press, Oxford (1993)

    MATH  Google Scholar 

  27. Ishibuchi, H., Nii, M.: Fuzzy regression using asymmetric fuzzy coefficients and fuzzified neural networks. Fuzzy Sets Syst. 119(2), 273–290 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lai, Y.-Jou, Hwang, C.-L.: Fuzzy mathematical programming. In: Fuzzy Mathematical Programming, pp. 74-186. Springer, Berlin, Heidelberg (1992)

  29. Linnemann, P.: van eeghen j., Loss reserving methods, surveys of actuarial studies no. 1. nationale-nederlanden n.v., rotterdam. 114 pages. ASTIN Bull. J. Int. Actuar. Assoc. 14(01), 87–88 (1984)

    Article  Google Scholar 

  30. Mack, T.: A simple parametric model for rating automobile insurance or estimating IBNR claims reserves. ASTIN Bull. 21(01), 93–109 (1991)

    Article  Google Scholar 

  31. Nielsen, J.P., Agbeko, T., Miranda, M.D.M., Verrall, R.J.: Validating the double chain ladder stochastic claims reserving model. Var. Adv. Sci. Risk 8(2), 138–160 (2014)

    Google Scholar 

  32. Party, C. T. F. W.: The estimation of loss development tail factors: a summary report. In: Casualty Actuarial Society Forum (2013)

  33. Straub, E., Swiss, A.A.: Non-life Insurance Mathematics. Springer, Berlin (1988)

    Book  MATH  Google Scholar 

  34. Taylor, G.: Claims Reserving in Non-life Insurance, Insurance Series. North-Holland, New York (1986)

    MATH  Google Scholar 

  35. Taylor, G., McGuire, G., Greenfield, A.: Loss reserving: past, present and future, vol. 102.  University of Melbourne, Research paper, ISBN 0 7340 2898 9 (2003)

  36. Van Eeghen, J.: Loss Reserving Methods, vol. 1. Research Department, Nationale-Nederlanden NV, The Hague (1981)

    Google Scholar 

  37. Wthrich, M.V., Merz, M.: Stochastic Claims Reserving Methods in Insurance, vol. 435. Wiley, Hoboken (2008)

    MATH  Google Scholar 

  38. Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by African Union through Pan African University/Jomo Kenyatta University of Agriculture and Technology. The authors acknowledge reviewers for their helpful comments.

Author information

Authors and Affiliations

  1. Institute of Basic Sciences Technology and Innovation, Pan African University-Jomo Kenyatta University of Agriculture and Technology, P.O. Box 62000-00200, Nairobi, Kenya

    Woundjiagué Apollinaire

  2. National Advanced School of Engineering, University of Maroua, P.O. Box 46, Maroua, Cameroon

    Woundjiagué Apollinaire

  3. Faculty of Science, University of Yaounde I, P.O. Box 812, Yaoundé, Cameroon

    Mbele Bidima Martin Le Doux

  4. School of Computing and Information Technology, Jomo Kenyatta University of Agriculture and Technology, P.O. Box 62000-00200, Nairobi, Kenya

    Waweru Mwangi Ronald

Authors
  1. Woundjiagué Apollinaire
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mbele Bidima Martin Le Doux
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Waweru Mwangi Ronald
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Woundjiagué Apollinaire.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Apollinaire, W., Le Doux, M.B.M. & Ronald, W.M. A Fuzzy Least-Squares Estimation of a Hybrid Log-Poisson Regression and Its Goodness of Fit for Optimal Loss Reserves in Insurance. Int. J. Fuzzy Syst. 21, 930–944 (2019). https://doi.org/10.1007/s40815-018-0564-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40815-018-0564-6

Keywords

Search

Navigation