Empirical Bayes sequential estimation for exponential families: The untruncated component

Abstract

We consider the empirical Bayes decision problem where the component problem is the sequential estimation of the mean θ of one-parameter exponential family of distributions with squared error loss for the estimation error and a cost c>0 for each observation. The present paper studies the untruncated sequential component case. In particular, an untruncated asymptotically pointwise optimal sequential procedure is employed as the component. With sequential components, an empirical Bayes decision procedure selects both a stopping time and a terminal decision rule for use in the component with parameter θ. The goodness of the empirical Bayes sequential procedure is measured by comparing the asymptotic behavior of its Bayes risk with that of the component procedure as the number of past data increases to infinity. Asymptotic risk equivalence of the proposed empirical Bayes sequential procedure to the component procedure is demonstrated.