Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T21:52:29.519Z Has data issue: false hasContentIssue false
  • English
  • Français

A Bootstrap Test for the Probability of Ruin in the Compound Poisson Risk Process

Published online by Cambridge University Press:  09 August 2013

Benjamin Baumgartner  and
Riccardo Gatto
Benjamin Baumgartner
Affiliation:
Institute of Mathematical Statistics and Actuarial Science, University of Bern, Alpeneggstrasse 22, 3012 Bern, Switzerland, E-mail: [email protected], E-mail: [email protected]
Riccardo Gatto
Affiliation:
Institute of Mathematical Statistics and Actuarial Science, University of Bern, Alpeneggstrasse 22, 3012 Bern, Switzerland, E-mail: [email protected], E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article we propose a bootstrap test for the probability of ruin in the compound Poisson risk process. We adopt the P-value approach, which leads to a more complete assessment of the underlying risk than the probability of ruin alone. We provide second-order accurate P-values for this testing problem and consider both parametric and nonparametric estimators of the individual claim amount distribution. Simulation studies show that the suggested bootstrap P-values are very accurate and outperform their analogues based on the asymptotic normal approximation.

Keywords

Edgeworth expansionexponential and log-normal claim amountsnormal approximationP-valuepivotal quantityresamplingsecond-order accuracy
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 40 , Issue 1 , May 2010 , pp. 241 - 255
Copyright
Copyright © International Actuarial Association 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Asmussen, S. (1985) Conjugate processes and the simulation of ruin problems, Stochastic Processes and their Applications, 20, 213–229.Google Scholar
Asmussen, S. (2000) Ruin Probabilities, vol. 2 of Advanced Series on Statistical Science and Applied Probability, World Scientific, Singapore.Google Scholar
Asmussen, S. and Rolski, T. (1991) Computational methods in risk theory: A matrix-algorithmic approach, Insurance: Mathematics and Economics, 10, 259–274.Google Scholar
Barndorff-Nielsen, O.E. and Schmidli, H.U. (1995) Saddlepoint approximations for the probability of ruin in finite time, Scandinavian Actuarial Journal, 2, 169–186.Google Scholar
Bowers, N., Gerber, H.U., Hickmann, J., Jones, D. and Nesbitt, C. (1997) Actuarial Mathematics, The Society of Actuaries, Schaumburg, Illinois, second edn.Google Scholar
Casella, G. and Berger, R.L. (2002) Statistical Inference, Duxbury Advanced Series in Statistics and Decision Sciences, Duxbury Press, Pacifice Grove, California, second edn.Google Scholar
Davison, A.C. and Hinkley, D.V. (2003) Bootstrap Methods and their Applications, no. 1 in Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, reprinted and corrected edn.Google Scholar
Dufresne, F. and Gerber, H.U. (1989) Three methods to calculate the probability of ruin, ASTIN Bulletin, 19,71–90.Google Scholar
Efron, B. (1979) Bootstrap methods: Another look at the jackknife, The Annals of Statistics, 7, 1–26.CrossRefGoogle Scholar
Efron, B. (1982) The Jackknife, the Bootstrap and Other Resampling Plans, no. 38 in CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia.CrossRefGoogle Scholar
Embrechts, P. and Mikosch, T. (1991) A bootstrap procedure for estimating the adjustment coefficient, Insurance: Mathematics and Economics, 10, 181–190.Google Scholar
Gatto, R. (2008) A saddlepoint approximation to the probability of ruin in the compound Poisson process with diffusion, Statistics and Probability Letters, 78, 1948–1954.CrossRefGoogle Scholar
Gerber, H.U. (1979) An Introduction to Mathematical Risk Theory, no. 8 in Huebner Foundation Monograph Series, S. S. Huebner Foundation for Insurance Education, Philadelphia.Google Scholar
Hall, P. (1992) The Bootstrap and Edgeworth Expansion, Springer Series in Statistics, Springer-Verlag, New York.CrossRefGoogle Scholar
Hipp, C. (1989a) Estimators and bootstrap confidence intervals for ruin probabilities, ASTIN Bulletin, 19, 57–70.Google Scholar
Jones, M.C. and Henderson, D.A. (2007) Kernel-type density estimation on the unit interval, Biometrika, 94, 977–984.CrossRefGoogle Scholar
Loisel, S., Mazza, C. and Rullière, D. (2008) Robustness analysis and convergence of empirical finite-time ruin probabilities and estimation risk solvency margin, Insurance: Mathematicsand Economics, 42, 746–762.Google Scholar
Pitts, S.M. (1994) Non-parametric estimation of compound distributions with applications in insurance, Annals of the Institute of Statistical Mathematics, 46, 537–555.CrossRefGoogle Scholar
Pitts, S.M., Grübel, R. and Embrechts, P. (1996) Confidence bounds for the adjustment coefficient, Advancesin Applied Probability, 28, 802–827.CrossRefGoogle Scholar
Politis, K. (2003) Semiparametric estimation for non-ruinprobabilities, Scandinavian Actuarial Journal, 1, 75–96.Google Scholar
Quenouille, M.H. (1949) Approximate tests of correlation in time-series, Journal of the Royal Statistical Society. Series B (Methodological), 11, 68–84.Google Scholar
R Development Core Team (2008) R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria.Google Scholar
Tukey, J.W. (1958) Bias and confidence in not quite largesamples (abstract), The Annals of Mathematical Statistics, 29, 614.Google Scholar