A simple predictive density based on the p*-formula

The predictive density proposed by Harris (1989) is based on integrating the density for a new observation with respect to the estimated sampling distribution of the maximum likelihood estimator 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 which consists of approximating the sampling distribution of the maximum likelihood estimator of the unknown parameter by Barndorff-Nielsen's (1983) p*-formula, and then using a Laplace approximation with O(n(-1)) correction terms for integrating out the parameter. The result can generally be expressed in terms of standard likelihood derivatives, and takes a quite simple form for exponential families and for location models.