Advertisement

Annals of the Institute of Statistical Mathematics

, Volume 46, Issue 3, pp 537–555 | Cite as

Nonparametric estimation of compound distributions with applications in insurance

  • S. M. Pitts
Estimation
  • 102 Downloads

Abstract

A nonparametric estimator of the distribution functionG of a random sum of independent identically distributed random variables, with distribution functionF, is proposed in the case where the distribution of the number of summands is known and a random sample fromF is available. This estimator is found by evaluating the functional that mapsF ontoG at the empirical distribution function based on the random sample. Strong consistency and asymptotic normality of the resulting estimator in a suitable function space are established using appropriate continuity and differentiability results for the functional. Bootstrap confidence bands are also obtained. Applications to the aggregate claims distribution function and to the probability of ruin in the Poisson risk model are presented.

Key words and phrases

Compound distributions nonparametric estimation aggregate claims probability of ruin 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Chow, Y. S. and Teicher, H. (1988).Probability Theory: Independence, Interchangeability, Martingales, 2nd ed., Springer, New York.Google Scholar
  2. Croux, K. and Veraverbeke, N. (1990). Nonparametric estimators for the probability of ruin,Insurance Math. Econom.,9, 127–130.Google Scholar
  3. Csörgő, S. and Teugels, J. L. (1990). Empirical Laplace transforms and approximation of compound distributions,J. Appl. Probab.,27, 88–101.Google Scholar
  4. Embrechts, P. and Veraverbeke, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims,Insurance Math. Econom.,1, 55–72.Google Scholar
  5. Embrechts, P., Jensen, J. L., Maejima, M. and Teugels, J. L. (1985a). Approximation for compound poisson and pólya processes,Adv. in Appl. Probab.,17, 623–637.Google Scholar
  6. Embrechts, P., Maejima, M. and Teugels, J. L. (1985b). Asymptotic behaviour of compound distributions,ASTIN Bulletin,15, 45–47.Google Scholar
  7. Frees, E. W. (1986a). Nonparametric estimation of the probability of ruin,ASTIN Bulletin,16S, 81–90.Google Scholar
  8. Frees, E. W. (1986b). Nonparametric renewal function estimation,Ann. Statist.,14, 1366–1378.Google Scholar
  9. Gill, R. D. (1989). Non- and semi-parametric maximum likelihood estimators and the von Mises method (Part 1),Scand. J. Statist.,16, 97–128.Google Scholar
  10. Grimmett, G. and Stirzaker, D. (1982).Probability and Random Processes, Clarendon Press, Oxford.Google Scholar
  11. Grübel, R. (1989). Stochastic models as functionals: Some remarks on the renewal case,J. Appl. Probab.,26, 296–303.Google Scholar
  12. Grübel, R. and Pitts, S. M. (1993). Nonparametric estimation in renewal theory I: The empirical renewal function,Ann. Statist. (to appear).Google Scholar
  13. Hipp, C. (1989). Estimators and bootstrap confidence intervals for ruin probabilities,ASTIN Bulletin,19, 57–69.Google Scholar
  14. Pollard, D. (1984).Convergence of Stochastic Processes, Springer, New York.Google Scholar
  15. Ramlau-Hansen, H. (1988). A solvency study in non-life insurance: Part 1. Analyses of fire, windstorm and glass claims,Scand. Actuar. J., 3–34.Google Scholar
  16. Shorack, G. R., and Wellner, J. A. (1986).Empirical Processes with Applications to Statistics, Wiley, New York.Google Scholar
  17. von Chossy, R. and Rappl, G. (1983). Some approximation methods for compound distributions,Insurance Math. Econom.,2, 251–270.Google Scholar
  18. Willekens, E. (1989). Asymptotic approximations of compound distributions and some applications,Bull. Soc. Math. Belg. Ser. B,41, 55–61.Google Scholar

Copyright information

© The Institute of Statistical Mathematics 1994

Authors and Affiliations

  • S. M. Pitts
    • 1
  1. 1.Department of Statistical ScienceUniversity College LondonLondonU.K.

Personalised recommendations