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

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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 +1, * 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 +1, * 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.