Efficient probabilistic forecasts for counts

Efficient probabilistic forecasts of integer-valued random variables are derived. The optimality is achieved by estimating the forecast distribution non-parametrically over a given broad model class and proving asymptotic (non-parametric) efficiency in that setting. The method is developed within the context of the integer auto-regressive class of models, which is a suitable class for any count data that can be interpreted as a queue, stock, birth-and-death process or branching process. The theoretical proofs of asymptotic efficiency are supplemented by simulation results that demonstrate the overall superiority of the non-parametric estimator relative to a misspecified parametric alternative, in large but finite samples. The method is applied to counts of stock market iceberg orders. A subsampling method is used to assess sampling variation in the full estimated forecast distribution and a proof of its validity is given.