Random function prediction and Stein's identity

Let X=(X1,X2,...,Xn) be a size n sample of i.i.d. random variables, whose distribution belong to the one-parameter ([theta]) continuous exponential family. We examine prediction functions of the form [theta]mh(X),m[greater-or-equal, slanted]1, where h is a polynomial in X. A natural identity that first appeared in Stein (Stein, 1973) and has been widely exploited since, is discussed in relation to members of such a family. Mild regularity conditions are also introduced that imply the nonexistence of a uniformly minimum mean squared error predictor for these functions.