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

A proof of the Cram\'er-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 $L^2$-differentiability conditions. It is also proved that if an efficient predictor (i.e. which risk attains the bound) exists then 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.