Advertisement

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

  • Woundjiagué Apollinaire
  • Mbele Bidima Martin Le Doux
  • Waweru Mwangi Ronald
Article
  • 7 Downloads

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.

Keywords

Fuzzy least-squares Log-Poisson Goodness of fit Loss reserve Hybrid 

Notes

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.

References

  1. 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)Google Scholar
  2. 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. 3.
    Asai, H.T.-S.U.-K.: Linear regression analysis with fuzzy model. IEEE Trans. Syst. Man Cybern. 12, 903–907 (1982)CrossRefGoogle Scholar
  4. 4.
    Barnett, G., Zehnwirth, B.: Best estimates for reserves. In: Proceedings of the Casualty Actuarial Society, vol. 87, No. 167, pp. 245–321 (2000)Google Scholar
  5. 5.
    Björkwall, S., Hssjer, O., Ohlsson, E.: Bootstrapping the separation method in claims reserving. ASTIN Bull. J. IAA 40(2), 845–869 (2010)MathSciNetzbMATHGoogle Scholar
  6. 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)Google Scholar
  7. 7.
    Bornhuetter, R.L., Ferguson, R.E.: The actuary and IBNR. In: Proceedings of the Casualty Actuarial Society, vol. 59, pp. 181–195 (1972)Google Scholar
  8. 8.
    Buckley, J.J.: Fuzzy Probability and Statistics, vol. 196. Springer, Berlin (2006)zbMATHGoogle Scholar
  9. 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)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Celmiš, A.: Least squares model fitting to fuzzy vector data. Fuzzy Sets Syst. 22(3), 245–269 (1987a)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Celmiš, A.: Multidimensional least-squares fitting of fuzzy models. Math. Model. 9(9), 669–690 (1987b)CrossRefGoogle Scholar
  12. 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)Google Scholar
  13. 13.
    Dahms, R.: Linear stochastic reserving methods. ASTIN Bull. J. IAA 42(1), 1–34 (2012)MathSciNetzbMATHGoogle Scholar
  14. 14.
    de Andrés Sánchez, J.: Calculating insurance claim reserves with fuzzy regression. Fuzzy Sets Syst. 157(23), 3091–3108 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    de Andrés-Sánchez, J.: Claim reserving with fuzzy regression and taylors geometric separation method. Insur. Math. Econ. 40(1), 145–163 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    de Andrés-Sánchez, J.: Claim reserving with fuzzy regression and the two ways of anova. Appl. Soft Comput. 12(8), 2435–2441 (2012)CrossRefGoogle Scholar
  17. 17.
    de Andrés Sánchez, J.: Fuzzy claim reserving in non-life insurance. Comput. Sci. Inf. Syst. 11(2), 825–838 (2014)CrossRefGoogle Scholar
  18. 18.
    de Campos Ibáñez, L.M., Muñoz, A.G.: A subjective approach for ranking fuzzy numbers. Fuzzy Sets Syst. 29(2), 145–153 (1989)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Dubois, D., Prade, H.: Operations on fuzzy numbers. Int. J. Syst. Sci. 9(6), 613–626 (1978)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Dubois, D., Prade, H.: Fuzzy numbers: an overview. In: Readings in Fuzzy Sets for Intelligent Systems, pp. 112–148. Morgan Kaufmann Publishers (1993)Google Scholar
  21. 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)MathSciNetCrossRefGoogle Scholar
  22. 22.
    D’Urso, P., Gastaldi, T.: A least-squares approach to fuzzy linear regression analysis. Comput. Stat. Data Anal. 34(4), 427–440 (2000)CrossRefGoogle Scholar
  23. 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)Google Scholar
  24. 24.
    England, P., Verrall, R.: Analytic and bootstrap estimates of prediction errors in claims reserving. Insur. Math. Econ. 25(3), 281–293 (1999)CrossRefGoogle Scholar
  25. 25.
    England, P.D., Verrall, R.J.: Stochastic claims reserving in general insurance. Br. Actuar. J. 8(03), 443–518 (2002)CrossRefGoogle Scholar
  26. 26.
    Francis, B., Green, M., Payne, C.: The GLIM System: Release 4 Manual. Clarendon Press, Oxford (1993)zbMATHGoogle Scholar
  27. 27.
    Ishibuchi, H., Nii, M.: Fuzzy regression using asymmetric fuzzy coefficients and fuzzified neural networks. Fuzzy Sets Syst. 119(2), 273–290 (2001)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Lai, Y.-Jou, Hwang, C.-L.: Fuzzy mathematical programming. In: Fuzzy Mathematical Programming, pp. 74-186. Springer, Berlin, Heidelberg (1992)Google Scholar
  29. 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)CrossRefGoogle Scholar
  30. 30.
    Mack, T.: A simple parametric model for rating automobile insurance or estimating IBNR claims reserves. ASTIN Bull. 21(01), 93–109 (1991)CrossRefGoogle Scholar
  31. 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. 32.
    Party, C. T. F. W.: The estimation of loss development tail factors: a summary report. In: Casualty Actuarial Society Forum (2013)Google Scholar
  33. 33.
    Straub, E., Swiss, A.A.: Non-life Insurance Mathematics. Springer, Berlin (1988)CrossRefGoogle Scholar
  34. 34.
    Taylor, G.: Claims Reserving in Non-life Insurance, Insurance Series. North-Holland, New York (1986)zbMATHGoogle Scholar
  35. 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)Google Scholar
  36. 36.
    Van Eeghen, J.: Loss Reserving Methods, vol. 1. Research Department, Nationale-Nederlanden NV, The Hague (1981)Google Scholar
  37. 37.
    Wthrich, M.V., Merz, M.: Stochastic Claims Reserving Methods in Insurance, vol. 435. Wiley, Hoboken (2008)zbMATHGoogle Scholar
  38. 38.
    Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)CrossRefGoogle Scholar

Copyright information

© Taiwan Fuzzy Systems Association and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Basic Sciences Technology and InnovationPan African University-Jomo Kenyatta University of Agriculture and TechnologyNairobiKenya
  2. 2.National Advanced School of EngineeringUniversity of MarouaMarouaCameroon
  3. 3.Faculty of ScienceUniversity of Yaounde IYaoundéCameroon
  4. 4.School of Computing and Information TechnologyJomo Kenyatta University of Agriculture and TechnologyNairobiKenya

Personalised recommendations