Sufficiency and efficiency in statistical prediction

This paper deals with existence and construction of optimal unbiased statistical predictors. Such predictors are functions of prediction-sufficient statistics. We first give a condition that links prediction-sufficiency in the observed model with sufficiency in the global model. We also show that existence of unbiased predictors does not imply existence of an optimal unbiased predictor.Finally, by using a Cramer Rao type inequality for predictors, we show that an efficient unbiased predictor does exist if and only if the model is exponential in some extended sense. Applications to autoregressive, Poisson and Ornstein–Uhlenbeck processes are considered.