Frequentist prediction intervals and predictive distributions

We consider parametric frameworks for the prediction of future values of a random variable Y, based on previously observed data X. Simple pivotal methods for obtaining calibrated prediction intervals are presented and illustrated. Frequentist predictive distributions are defined as confidence distributions, and their utility is demonstrated. A simple pivotal-based approach that produces prediction intervals and predictive distributions with well-calibrated frequentist probability interpretations is introduced, and efficient simulation methods for producing predictive distributions are considered. Properties related to an average Kullback-Leibler measure of goodness for predictive or estimated distributions are given. The predictive distributions here are shown to be optimal in certain settings with invariance structure, and to dominate plug-in distributions under certain conditions.