Credibility Approximations for Bayesian Prediction of Second Moments.
W. S. Jewell, R. Schnieper · 1984
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.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 x 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.