On optimal estimation with respect to a large family of cost functions

The authors consider the problem of optimal estimation of a random variable X based on an observation denoted by a random vector Y. A commonly encountered problem involves estimating X via h(Y) so as to minimize E( Phi (X-h(Y))), where h is Borel measurable and Phi is a Borel measurable cost function chosen to adequately reflect the fidelity demands of the problem under consideration. The authors place a mild condition on the regular conditional distribution of X given sigma (Y) that ensures that E( Phi (X-h(Y))) is minimized for any cost function Phi that is nonnegative, even and convex. In addition, it is shown that given any Borel measurable function g: R to R, there exist random variables X and Y possessing a joint density function such that E(X mod Y=y)=g(y) almost everywhere with respect to Lebesgue measure.