An Asymptotic Second-Order Lower Bound for the Bayes Risk of a Sequential Procedure

Abstract

Consider the problem of estimating the difference of the means of two populations, where each population distribution is a member of the one-parameter exponential family of probability distributions. A Bayesian approach is adopted in which the mean difference is estimated under the squared error loss and the prior distributions are of the form proposed by Diaconis and Ylivisaker [Diaconis, P.; Ylivisaker, D. Conjugate priors for exponential families. Ann. of Statist. 1979, 6, 269–281]. The main result determines an asymptotic second-order lower bound for the Bayes risk of a sequential procedure that takes N observations from one population and t − N from the other population, and estimates the mean difference by the Bayes estimator, where N is determined according to a sequential design and t denotes the total number of observations sampled from both populations.

Key Words:

• Bayes estimator
• Conjugate prior distribution
• Fatou's lemma
• Sequential design
• Squared error loss
• Uniform integrability

Mathematics Subject Classification:

• Primary 62L05
• Secondary 62L10

Acknowledgments

We are very pleased to thank Professor Nitis Mukhopadhyay, the associate editor, and the referee for many helpful suggestions and remarks on an earlier version of this paper.

Notes

Recommended by B. Boukai