Prediction functions and geostatistics

We consider analogues of estimating functions for situations in which the prediction of observables is of primary interest. We show that the mode of the predictive density has an optimality property for prediction analogous to a similar optimality property for the mode of the posterior density in the case of parametric estimation. Applications of predictive estimating functions in spatial statistics with particular reference to the geostatistical method known as kriging are developed.