On empirical Bayes two-tail tests in a positive exponential family

摘要

This paper deals with a two-tail testing problem for the parameter θ in a positive exponential family with probability density f(x|θ) = u(x)η(θ) exp(—x/θ) using the empirical Bayes approach. Based on the accumulated historical data, an empirical Bayes test δ_n~* is constructed by mimicking the behavior of a Bayes test. The empirical Bayes test δ_n~* is shown to be asymptotically optimal. The rates of convergence associated with the regret of δ_n~* is investigated and shown to be either ln~3 n or n~(-3/4) ln~2 n according to the pertaining conditions, where n is the number of past data available when the current decision problem is considered.