Nonparametric Sequential Bayes Estimation of the Distribution Function

Abstract

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, 1967 1968 1969a b) 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.

Recommended by Michael Baron

Keywords:

• Asymptotically pointwise optimal (APO)
• Bayes stopping rule
• Dirichlet process
• Squared error loss

Mathematics Subject Classification:

• 62L12
• 62L15
• 62G05

ACKNOWLEDGMENTS

We thank the editor, the associate editor, and the referees for their comments, which significantly improved the content and presentation of the paper. Research of Malay Ghosh was partially supported by NSF Grant Number SES-0317589.

Notes

Recommended by Michael Baron