Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T15:14:16.055Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonparametric Estimation of the Probability of Ruin

Published online by Cambridge University Press:  29 August 2014

Edward W. Frees
Edward W. Frees*
Affiliation:
University of Wisconsin-Madison
*
School of Business, University of Wisconsin-Madison, 1155 Observatory Drive, Madison, WI53706, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The finite and infinite horizon time probability of ruin are important parameters in the study of actuarial risk theory. This paper introduces procedures for directly estimating these key parameters from a random sample of observations without assumptions as to the parametric form of the distribution from which the observations arise. The estimators introduced apply to most of the classical models in which ruin probabilities are used and also apply to a much broader class of models. The procedures are based on the concept of sample reuse, an old idea in statistics which is becoming more popular due to the widespread availability of high speed computers. In this paper, the almost sure consistency of the estimators is established. Further, finite sample properties of the estimators are investigated in a simulation study.

Keywords

Reverse martingalesample reuse
Type
Astin Competition 1985: Prize-Winning Papers and Other Selected Papers
Information
ASTIN Bulletin: The Journal of the IAA , Volume 16 , Issue S1 , April 1986 , pp. S81 - S90
Copyright
Copyright © International Actuarial Association 1986

References

Asmussen, S. (1984) Approximations for the Probability of Ruin within Finite Time. Scandinavian Actuarial Journal 31–57.CrossRefGoogle Scholar
Beard, R., Pentikäinen, T. and Pesonen, E. (1984) Risk Theory: The Stochastic Basis of Insurance. Chapman and Hall: New York.CrossRefGoogle Scholar
Bühlmann, H. (1970) Mathematical Methods in Risk Theory. Springer-Verlag: Heidelberg.Google Scholar
Csörgö, M. and Révész, P. (1981) Strong Approximations in Probability and Statistics. Academic Press: New York.Google Scholar
Efron, B. (1982) The Jackknife, the Bootstrap and Other Resampling Plans. SIAM: Philadelphia.CrossRefGoogle Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Applications, Vol. 2. Wiley: New York.Google Scholar
Gerber, H. (1979) An Introduction to Mathematical Risk Theory. Irwin: Homewood, Illinois.Google Scholar
Hoeffding, W. (1948) A Class of Statistics with Asymptotically Normal Distributions. Annals of Mathematical Statistics 19, 293–325.CrossRefGoogle Scholar
Lalley, S. (1984) Limit Theorems for First-Passage Times in Linear and Non-Linear Renewal Theory. Advances in Applied Probability 16, 766–803.CrossRefGoogle Scholar
Seal, H. (1978) Survival Probabilities. Wiley: New York.Google Scholar
Seal, H. and Gerber, H. (1984) Mixed Poisson Processes and the Probability of Ruin. Insurance: Mathematics and Economics 3, 189–190.Google Scholar
Siegmund, D. (1979) Corrected Diffusion Approximations in Certain Random Walk Problems. Advances in Applied Probability 11, 701–719.CrossRefGoogle Scholar
Thorin, O. and Wikstad, N. (1976) Calculation of Ruin Probabilities When the Claim Distribution is Lognormal. Astin Bulletin 9, 231–246.CrossRefGoogle Scholar