Empirical Bayes testing for equivalence

For a fixed point θ 0 and a positive value c0, this paper studies the problem of testing the hypotheses H 0 : | θ − θ 0 | ≤ c 0 against H 1 : | θ − θ 0 | > c 0 for the normal mean parameter θ using the empirical Bayes approach. With the accumulated past data, a monotone empirical Bayes test is constructed by mimicking the behavior of a monotone Bayes test. Such an empirical Bayes test is shown to be asymptotically optimal and its regret converges to zero at a rate ( ln n ) 2.5 / n where n is the number of past data available, when the current testing problem is considered. A simulation study is also given, and the results show that the proposed empirical Bayes procedure has good performance for small to moderately large sample sizes. Our proposed method can be applied for testing close to a control problem or testing the therapeutic equivalence of one standard treatment compared to another in clinical trials.