Asymptotically pointwise optimal allocation rules in Bayes sequential estimation

In this paper, the concept of asymptotic pointwise optimality in a single sequence of random variables provided by Bickel and Yahav [Bickel, P.J., Yahav, J.A., 1967. Asymptotically pointwise optimal procedures in sequential analysis. In: Proc Fifth Berkeley Symp. Math Statist. Prob. 1. University of California Press, pp. 401-413] is extended to more than one sequence of random variables. The Bayesian sequential estimation problem for one-parameter exponential families is considered and an asymptotically pointwise optimal rule, which includes a sequential allocation procedure and a stopping time, is provided. Some properties of asymptotic optimality are also obtained for the rules with and without using the prior information.