Efficient prediction in $L^2$-differentiable families of distributions

Abstract

A proof of the Cramér-Rao inequality for prediction is presented under conditions of L 2 -differentiability of the family of distributions of the model. The assumptions and the proof differ from those of Miyata ( 2001) who also proved this inequality under L 2differentiability conditions. It is also proved that an efficient predictor (i.e. which risk attains the bound) exists if and only if the family of distributions is of a special form which can be seen as an extension of the notion of exponential family. This result is also proved under L 2 -differentiability conditions.