Summary: [For the entire collection see Zbl 0702.00028.]
Hierarchical models are important in Bayesian prediction because they enable the use of collateral data from related risks with exchangeable parameters. The D. V. Lindley and A. F. M. Smith [J. roy. statist. Soc., Ser. B 34, 1-41 (1972; Zbl 0246.62050)] normal-normal- normal model with random means shows clearly how the linear predictive mean for a single risk is improved by the availability of cohort data. However, this model has the disadvantage that the predictive density is homoscedastic, that is, the posterior variance depends only on the design (number of risks and number of samples). In most applications, one would assume that the variance also depended upon the data values. One can change the variances at each level into random parameters, but this modifies the predictive mean formulae and leads to messy results in general.
This paper describes an extended normal model with variances that are quadratic in the data, and with the additional advantage that the linear mean formulae are unchanged.