Credibility Approximations for Bayesian Prediction of Second Moments.
Abstract
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. Buhlmann has given a credibility approximation to the variance in terms of the sample mean and sample variance. This paper presents 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 x 3 linear system, using 11 prior moments from the collective and giving joint predictions of all moments of interest. Previously developed special class follow immediately. For many analytic models of interest, one can replace the 3-dimensional joint prediction with three independent credibility forecasts using the natural statistics for each moment. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Mar 01, 1984
- Accession Number
- ADA144996
Entities
People
- R. Schnieper
- W. S. Jewell
Organizations
- University of California, Berkeley