Improved prediction limits for a general class of Gaussian models

In this article we consider the problem of prediction for a general class of Gaussian models, which includes, among others, autoregressive moving average time-series models, linear Gaussian state space models and Gaussian Markov random fields. Using an idea presented in Sjöstedt-De Luna and Young (2003), in the context of spatial statistics, we discuss a method for obtaining prediction limits for a future random variable of interest, taking into account the uncertainty introduced by estimating the unknown parameters. The proposed prediction limits can be viewed as a modification of the estimative prediction limit, with unconditional, and eventually conditional, coverage error of smaller asymptotic order. The modifying term has a quite simple form and it involves the bias and the mean square error of the plug-in estimators for the conditional expectation and the conditional variance of the future observation. Applications of the results to Gaussian time-series models are presented.