On the Asymptotic Expansion of the Mean Square Risk for a Bayesian Estimate

We consider the problem of estimation of an unknown parameter affected by unit Gaussian noise. We show that continuity of the a priori density $p(x)$ of this parameter and mean-square differentiability of $\sqrt{p(x)}$ are necessary and sufficient conditions for the second term of the asymptotic expansion of the mean square risk of a Bayesian estimate to be of order $n^{-2}$.