A discussion of parameter and model uncertainty in insurance. (English) Zbl 0971.62063

Summary: We consider the process of modelling uncertainty. In particular, we are concerned with making inferences about some quantity of interest which, at present, has been unobserved. Examples of such a quantity include the probability of ruin of a surplus process, the accumulation of an investment, the level or surplus or deficit in a pension fund and the future volume of new business in an insurance company.
Uncertainty in this quantity of interest, \(y\), arises from three sources: (1) uncertainty due to the stochastic nature of a given model; (2) uncertainty in the values of the parameters in a given model; (3) uncertainty in the model underlying what we are able to observe and determining the quantity of interest. It is common in actuarial science to find that the first source of uncertainty is the only one which receives rigorous attention. A limited amount of research in recent years has considered the effect of parameter uncertainty, while there is still considerable scope for development of methods which deal in a balanced way with model risk. Here we discuss a methodology which allows all three sources of uncertainty to be assessed in a more coherent fashion.


62P05 Applications of statistics to actuarial sciences and financial mathematics
62F15 Bayesian inference
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[1] Bawa, V., Brown, S., Klein, R. (Eds.), 1979. Estimation Risk and Optimal Portfolio Choice. North-Holland, Amsterdam. · Zbl 0425.90001
[2] Bernardo, J.M., Smith, A.F.M., 1994. Bayesian Theory. Wiley, Chichester, UK.
[3] Chatfield, C., Model uncertainty, data mining and statistical inference (with discussion), Journal of the royal statistical society series A, 158, 419-466, (1995)
[4] Clemen, R.T., Combining forecasts: a review and annotated bibliography, International journal of forecasting, 5, 559-583, (1989)
[5] Crouhy, M.; Galai, D.; Mark, R., Model risk, Journal of financial engineering, 7, 267-288, (1998)
[6] Dawid, A.P., Invariant prior distributions, Encyclopaedia of statistical sciences, 4, 228-236, (1983)
[7] Daykin, C.D., Pentikäinen, T., Pesonen, M., 1994. Practical Risk Theory for Actuaries. Chapman & Hall, London. · Zbl 1140.62345
[8] Dickson, D.C.M.; Tedesco, L.M.; Zehnwirth, B., Predictive aggregate claims distributions, Journal of risk and insurance, 65, 689-709, (1998)
[9] Draper, D., Assessment and propagation of model uncertainty (with discussion), Journal of the royal statistical society series B, 57, 45-97, (1995) · Zbl 0812.62001
[10] Gilks, W.R., Richardson, S., Spiegelhalter, D.J., 1995. Markov Chain Monte Carlo in Practice. Chapman & Hall, London. · Zbl 0832.00018
[11] Harris, G.R., Markov chain Monte Carlo estimation of regime switching vector autoregressions, ASTIN bulletin, 29, 47-79, (1999) · Zbl 1162.91524
[12] Jeffreys, H., 1961. Theory of Probability, 3rd Edition. Oxford University Press, Oxford. · Zbl 0116.34904
[13] Kass, R.E.; Raftery, A.E., Bayes factors, Journal of the American statistical association, 90, 773-795, (1995) · Zbl 0846.62028
[14] Klugman, S., 1992. Bayesian Statistics in Actuarial Science. Kluwer Academic Publishers, Boston, MA. · Zbl 0753.62075
[15] McNeil, A.J., Estimating the tails of loss severity distributions using extreme value theory, ASTIN bulletin, 27, 117-137, (1997)
[16] Min, C.-K.; Zellner, A., Bayesian and non-Bayesian methods for combining models and forecasts with applications to forecasting international growth rates, Journal of econometrics, 56, 89-118, (1993) · Zbl 0800.62800
[17] O’Hagan, A., 1994. Kendall’s Advanced Theory of Statistics. Vol. 2B. Bayesian Inference, Halstead Press, London.
[18] O’Hagan, A., Fractional Bayes factors for model comparison (with discussion), Journal of the royal statistical society series B, 57, 99-118, (1995) · Zbl 0813.62026
[19] Pai, J., Bayesian analysis of compound loss distributions, Journal of econometrics, 79, 129-146, (1997) · Zbl 0873.62117
[20] Panjer, H.H., Willmot, G.E., 1992. Insurance Risk Models. Society of Actuaries, Schaumburg.
[21] Parker, G., Stochastic analysis of the interaction between investment and insurance risks (with discussion), North American actuarial journal, 1, 55-84, (1997) · Zbl 1080.91530
[22] Schmock, U., Estimating the value of the wincat coupons of the winterthur insurance convertible bond, ASTIN bulletin, 29, 101-163, (1999) · Zbl 1162.91433
[23] Schwarz, G., Estimating the dimension of a model, Annals of statistics, 6, 461-464, (1978) · Zbl 0379.62005
[24] Scollnik, D.P.M., On the analysis of the truncated generalised Poisson distribution using a Bayesian method, ASTIN bulletin, 28, 135-152, (1998) · Zbl 1168.60311
[25] Shibata, R., Selection of the order of an autoregressive model by akaike’s information criterion, Biometrika, 63, 117-126, (1976) · Zbl 0358.62048
[26] Wei, W.W.S., 1990. Time Series Analysis. Addison-Wesley, Redwood City.
[27] Wilkie, A.D., More on a stochastic asset model for actuarial use (with discussion), British actuarial journal, 1, 777-964, (1995)
[28] Williams, D., 1991. Probability with Martingales. Cambridge University Press, Cambridge. · Zbl 0722.60001
[29] Zellner, A., Bayesian analysis of regression error terms, Journal of the American statistical association, 70, 138-144, (1975) · Zbl 0313.62050
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