Empirical Bayes with a changing prior (Q5896395)

Let \(\{(\theta_ i,X_ i)\}\), \(i=1,2,...\), be a sequence of independent random vectors with \(\theta_ i\) and \(X_ i\) real valued. Assume \(X_ i\) has a density belonging to an exponential family with natural parameter \(\theta_ i\) where \(X_ i\) is the natural observation and that \(\theta_ i\) has unknown distribution \(G^{(i)}\) with support independent of i. At the nth stage, for \(n=1,2,...\), having observed \(X_ 1,...,X_ n\), a decision must be made concerning \(\theta_ n\). A sequence of decision rules is asymptotically optimal if the difference between the risk in the nth problem and the Bayes risk for the prior \(G^{(n)}\) converges to 0 as \(n\to \infty.\) The authors show asymptotic optimality, giving rates of convergence, in two problems: (i) a sequence of hypothesis testing problems about the \(\theta_ i\), and (ii) estimation of the \(\theta_ i\) with squared error loss.