Empirical Bayes testing for success probability of Bernoulli process with negative binomial sampling: nonidentical components case

摘要

We study an empirical Bayes testing problem for the success probability of Bernoulli process with negative binomial sampling under nonidentical components situation. An empirical Bayes test δ n+1,n * is constructed based on an estimator c n of the critical point c G of the Bayes test δ n+1,G . The empirical Bayes test δ n+1,n * possesses the asymptotic optimality, and its associated regret converges to zero at an exponential decay rate O(exp (?n α)) for some positive value α, depending on the unknown prior distribution G. This result extends the exponential type decay rate of convergence obtained in Liang (Ann. Stat. 16:1635–1642, 1988; Stat. Probab. Lett. 44:241–249, 1999) to the nonidentical components case.