In an earlier note, the linear regression of a random variable Y on a (possibly vector) variable X, was shown to be the optimal function of X for predicting Y, if all that is known about the joint distribution of Y and X is their (joint and marginal) moments of orders one and two. The sense in which it is optimal, is that it is the unique minimax strategy for the statistician, for squared-error loss, if 'nature' can choose any joint distribution with the given moments. A. Kagan raised the question, whether quadratic regression would similarly be optimal, if the moments of orders 1 through 4 are given. A suitable generalization of this question is answered here in the affirmative.
