Bayes and empirical Bayes estimation with errors in variables

Suppose that the random variable X is distributed according to exponential families of distributions, conditional on the parameter [theta]. Assume that the parameter [theta] has a (prior) distribution G. Because of the measurement error, we can only observe Y = X + [var epsilon], where the measurement error [theta] is independent of X and has a known distribution. This paper considers the squared error loss estimation problem of [theta] based on the contaminated observation Y. We obtain an expression for the Bayes estimator when the prior G is known. For the case G is completely unknown, an empirical Bayes estimator is proposed based on a sequence of observations Y1, Y2,...,Yn, where Yi's are i.i.d. according to the marginal distribution of Y. It is shown that the proposed empirical Bayes estimator is asymptotically optimal.