In sequential analysis, Bayes stopping rules are often difficult to determine explicitly. Bickel and Yahav (1967, Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability, VI, pp 401 413) provided an attractive large sample approximation to sequential Bayes rules which they called "asymptotically pointwise optimal" (A.P.O.) rules. The present paper proposes A.P.O. rules for certain hierarchical and empirical Bayes models. These rules are shown to be asymptotically "non deficient" in the sense of Woodroofe (1981, Zeitschrift fiir Wahrscheinlich keitstheorie und Verwandte Gebiete, pp 331 341).