A simple predictive density based on the p*-formula

The predictive density proposed by Harris (1989) is based onintegrating the density for a new observation with respect tothe estimated sampling distribution of the maximum likelihoodestimator of the unknown parameter. This has good properties,but is rather complicated to compute even for simple models.An approximation to the Harris proposal is considered whichconsists of approximating the sampling distribution of the maximumlikelihood estimator of the unknown parameter by Barndorff-Nielsen's(1983) p*-formula, and then using a Laplace approximation withO(n^–1) correction terms for integrating out the parameter.The result can generally be expressed in terms of standard likelihoodderivatives, and takes a quite simple form for exponential familiesand for location models.