The problem of Bayes sequential point estimation without using the prior information is considered. Two sequential procedures of the type of Bickel and Yahav (1967, 1968), for the scale exponential and location normal families respectively, are proposed. It is shown in the present paper that the proposed procedures are asymptotically pointwise optimal (A.P.O.) in the sense of Bickel and Yahav (1967) and the second order approximation of the Bayes risk is established for a large class of prior distributions.