Bayes and empirical Bayes estimation with errors in variables

Abstract

Suppose that the random variable X is distributed according to exponential families of distributions, conditional on the parameter θ. Assume that the parameter θ has a (prior) distribution G. Because of the measurement error, we can only observe Y = X + ε, where the measurement error θ is independent of X and has a known distribution. This paper considers the squared error loss estimation problem of θ 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.