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 a has a (prior) distribution G. Because of the measurement error, we can only observe Y = X+epsilon, where the measurement error epsilon is independent of X and has a known distribution. This paper considers the squared error loss estimation problem of a 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 Y-1,Y-2,...,Y-n, where Y-i's are i.i.d. according to the marginal distribution of Y. It is shown that the proposed empirical Bayes estimator is asymptotically optimal.