Consistency of the Empirical Bayesian Analogue of the Regression Estimation

We study the possibility of constructing a Bayesian estimate based on a kernel estimate of the unconditional distribution density. We consider the situation when the observed random variable is the sum of an unknown parameter and a centered normal error with a known variance. In this case, the Bayesian estimate can be represented through the unconditional density of observations and its derivative, which makes it possible to construct empirical analogues of the Bayesian estimate only on the basis of density estimates. The consistency of these analogues is shown both for a fixed result of the current experiment, and in the mean.