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Parameter Uncertainty in Exponential Family Tail Estimation

Published online by Cambridge University Press:  09 August 2013

Z. Landsman  and
A. Tsanakas
Z. Landsman
Affiliation:
Department of Statistics, University of Haifa
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Abstract

Actuaries are often faced with the task of estimating tails of loss distributions from just a few observations. Thus estimates of tail probabilities (reinsurance prices) and percentiles (solvency capital requirements) are typically subject to substantial parameter uncertainty. We study the bias and MSE of estimators of tail probabilities and percentiles, with focus on 1-parameter exponential families. Using asymptotic arguments it is shown that tail estimates are subject to significant positive bias. Moreover, the use of bootstrap predictive distributions, which has been proposed in the actuarial literature as a way of addressing parameter uncertainty, is seen to double the estimation bias. A bias corrected estimator is thus proposed. It is then shown that the MSE of the MLE, the parametric bootstrap and the bias corrected estimators only differ in terms of order O(n−2), which provides decision-makers with some flexibility as to which estimator to use. The accuracy of asymptotic methods, even for small samples, is demonstrated exactly for the exponential and related distributions, while other 1-parameter distributions are considered in a simulation study. We argue that the presence of positive bias may be desirable in solvency capital calculations, though not necessarily in pricing problems.

Keywords

Reinsurance pricingVaRparameter uncertaintybiasbootstrapexponential families
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 42 , Issue 1 , May 2012 , pp. 123 - 152
Copyright
Copyright © International Actuarial Association 2012

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References

Borowicz, J.M. and Norman, J.P. (2008) The effects of parameter uncertainty in the extreme event frequency-severity model. Presented at the 28th International Congress of Actuaries, Paris, 28 May - 2 June 2008. Available at: www.ica2006.com/Papiers/3020/3020.pdf, Accessed April 12, 2011.Google Scholar
Cairns, A.J.G. (2000) A discussion of parameter and model uncertainty in insurance. Insurance: Mathematics and Economics 27, 313–30.Google Scholar
Cox, D.R. and Hinkley, D.V. (1979) Theoretical Statistics. Chapman and Hall.Google Scholar
Cummins, J.D. and Lewis, C.M. (2003) Catastrophic events, parameter uncertainty and the breakdown of implicit long-term contracting: The case of terrorism insurance. Journal of Risk and Uncertainty 26(2), 153–77.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and Its Applications: Vol. 2. Wiley.Google Scholar
Gerrard, R.J. and Tsanakas, A. (2010) Failure probability under parameter uncertainty. Risk Analysis, forthcoming.Google Scholar
Gradshteyn, I.S. and Ryzhik, I.M. (2007) Table of Integrals, Series, and Products. 7th ed. Academic Press.Google Scholar
Hall, P. (1997) The Bootstrap and Edgeworth Expansion. Springer.Google Scholar
Harris, I.R. (1989) Predictive fit for natural exponential families. Biometrika, 76(4), 675–84.Google Scholar
Klugman, S.A., Panjer, H.H. and Willmott, G.E. (2004) Loss Models: From Data to Decisions. Wiley.Google Scholar
Landsman, Z. (2004) Second order Bayes prediction of functionals of exponential dispersion distributions and an application to the prediction of the tails. ASTIN Bulletin 34(2), 285–298.Google Scholar
Mata, A. (2000) Parameter uncertainty for extreme value distributions. Presented at 2000 General Insurance Convension, 25-28 October. Available at: http://www.actuaries.org.uk/sites/all/files/documents/pdf/0151-0173.pdf, Accessed April 12, 2011.Google Scholar
Murray, J.D. (1974) Asymptotic Analysis. Clarendon Press.Google Scholar
Powers, M.R., Venezian, E.C. and Jucá, I.B. (2003) Of happy and hapless regulators: the asymptotics of ruin. Insurance: Mathematics and Economics, 32, 317–30.Google Scholar
Richards, S. (2009) Investigating parameter risk for Solvency II and ICAS. Life and Pensions Jun: 36–40.Google Scholar
Saltzmann, R. and Wuethrich, M.V. (2010) Cost-of-capital margin for a general insurance liability runoff. Astin Bulletin, 40(2), 415–451.Google Scholar
Smith, R.L. (1998) Bayesian and frequentist approaches to parametric predictive inference. In Bernardo, J.M., Berger, J.O., Dawid, A.P. and Smith, A.F.M. (eds). Bayesian Statistics 6. Oxford University Press.Google Scholar
Verrall, R.J. and England, P.D. (2006) Predictive distributions of outstanding liabilities in general insurance. Annals of Actuarial Science 1(2), 221–70.Google Scholar