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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

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