Nonparametric Sequential Bayes Estimation of the Distribution Function

The paper considers nonparametric estimation of the distribution function under weighted squared error loss plus cost. Sequential Bayes rules are proposed under Dirichlet process priors. Since the implementation of Bayes procedures via backward induction (cf. Arrow et al., 1949) is computationally prohibitive, we consider in this paper approximations to sequential Bayes rules. Like other problems of this type, it turns out, in this case, that asymptotically pointwise optimal (APO) stopping rules (cf. Bickel and Yahav, 196719681969ab) are equivalent to the myopic or one-step look-ahead rules. Also, we provide a proof of the first-order optimality and second-order expansion of the APO rules.