On an Aspect of Optimal Nonlinear Estimation

This paper considers the problem of optimal estimation of a random variable X based on an observation denoted by a random vector Y. A mild restriction on the regular conditional distribution function of X given o(Y) is presented which ensures that E[Q,(X-g(Y))] is minimized for any cost function Q, that is nonnegative, even, and convex. Further, we show that given any real valued Bore1 measurable function there exist random variables X and Y, possessing a joint density function, so that the chosen function is the optimal estimatm, with respect to any nonnegative, even, convex cost function Q,, of the random variable X as a function of the random variable Y. Finally, the estimation of X via a random variable measurable with respect to a given o-subalgebra is treated.