  The focus of this paper is on the quantification of sampling variation in
frequentist probabilistic forecasts. We propose a method of constructing
confidence sets that respects the functional nature of the forecast
distribution, and use animated graphics to visualize the impact of parameter
uncertainty on the location, dispersion and shape of the distribution. The
confidence sets are derived via the inversion of a Wald test and are
asymptotically uniformly most accurate and, hence, optimal in this sense. A
wide range of linear and non-linear time series models - encompassing long
memory, state space and mixture specifications - is used to demonstrate the
procedure, based on artificially generated data. An empirical example in which
distributional forecasts of both financial returns and its stochastic
volatility are produced is then used to illustrate the practical importance of
accommodating sampling variation in the manner proposed.
