Prediction of an unobserved random variable is considered from a frequentist viewpoint. After a brief review of previous work, a number of examples in which an exact solution is possible are given, partly for their intrinsic interest and partly to illustrate general results. A new form of predictive density is derived accurate to the third order of asymptotic theory under ordinary repeated sampling. The formula is invariant under transformation of the observed and unobserved random variables and under reparametrization. It respects the conditionality principle and may be based on the minimal prediction sufficient statistic. Some open problems are noted.
"Prediction and asymptotics." Bernoulli 2 (4) 319 - 340, December 1996.