Random function prediction and Stein's identity

Abstract

Let X =(X 1,X 2,…,X n) be a size n sample of i.i.d. random variables, whose distribution belong to the one-parameter ( θ) continuous exponential family. We examine prediction functions of the form θ mh( X ),m⩾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.