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Credibility Approximations for Bayesian Prediction of Second Moments

Published online by Cambridge University Press:  29 August 2014

William S. Jewell  and
Rene Schnieper
William S. Jewell
Affiliation:
University of California, Berkeley, California
Rene Schnieper
Affiliation:
Department of Mathematics, ETH-Zentrum, Zurich
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Abstract

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Credibility theory refers to the use of linear least-squares theory to approximate the Bayesian forecast of the mean of a future observation; families are known where the credibility formula is exact Bayesian. Second-moment forecasts are also of interest, for example, in assessing the precision of the mean estimate. For some of these same families, the second-moment forecast is exact in linear and quadratic functions of the sample mean. On the other hand, for the normal distribution with normal-gamma prior on the mean and variance, the exact forecast of the variance is a linear function of the sample variance and the squared deviation of the sample mean from the prior mean. Bühlmann has given a credibility approximation to the variance in terms of the sample mean and sample variance.

In this paper, we present a unified approach to estimating both first and second moments of future observations using linear functions of the sample mean and two sample second moments; the resulting least-squares analysis requires the solution of a 3 × 3 linear system, using 11 prior moments from the collective and giving joint predictions of all moments of interest. Previously developed special cases follow immediately. For many analytic models of interest, 3-dimensional joint prediction is significantly better than independent forecasts using the “natural” statistics for each moment when the number of samples is small. However, the expected squared-errors of the forecasts become comparable as the sample size increases.

Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 15 , Issue 2 , November 1985 , pp. 103 - 121
Copyright
Copyright © International Actuarial Association 1985

References

Bühlmann, H. (1967) Experience Rating and Credibility. Astin Bulletin 4, 199–207.CrossRefGoogle Scholar
Bühlmann, H. (1970) Mathematical Methods in Risk Theory. Springer-Verlag: New York.Google Scholar
Gerber, H. (1980) Introduction to the Mathematical Theory of Risk, The Huebner Foundation, Philadelphia, PA.Google Scholar
Hachemeister, C. A. (1974) Credibility for Regression Models with Application to Trend. In Credibility Theory and Applications, Kahn, P. M. (ed.), Academic Press: New York.Google Scholar
Jewell, W. S. (1974 a) Credible Means are Exact Bayesian for Simple Exponential Families. Astin Bulletin 7, 237–269.CrossRefGoogle Scholar
Jewell, W. S. (1974 b) Exact Multidimensional Credibility. Mitteilungen der Vereinigung schweizerischer Versicherungsmathematiker 74, 193–214.Google Scholar
Jewell, W. S. (1975) Regularity Conditions for Exact Credibility. Astin Bulletin 8, 336–341.CrossRefGoogle Scholar
Jewell, W. S. (1980) Models in Insurance: Paradigms, Puzzles, Communications, and Revolutions. Transactions 21st International Congress of Actuaries, Zürich and Lausanne, Volume S, 87–141.Google Scholar
Jewell, W. S. (1983) Enriched Multinormal Priors Revisited. Journal of Econometrics 23, 5–35.CrossRefGoogle Scholar
Morris, C. N. (1982) Natural Exponential Families with Quadratic Variance Functions. The Annals of Statistics 10, 65–80.CrossRefGoogle Scholar
Norberg, R. (1979) The Credibility Approach to Experience Rating. Scandinavian Actuarial Journal, 181–221.CrossRefGoogle Scholar