This paper develops asymptotic prediction functions that approximate the shape of the density of future observations and correct for parameter uncertainty. The functions are based on extensions to a definition of predictive likelihood originally suggested by Lauritzen and Hinkley. The prediction function is shown to possess efficiency properties based on the Kullback-Leibler measure of information loss. Examples of the application of the prediction function and the derivation of relative efficiency are shown for linear normal models, nonnormal models, and ARCH models.