Empirical Bayes testing for a normal mean: variance unknown case

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This paper deals with the empirical Bayes testing for the mean θ of a N ( θ , σ 2 ) distribution using a linear error loss where it is assumed that θ follows an unknown prior distribution G and variance σ 2 is fixed but unknown. An empirical Bayes test δ ˜ n is constructed. Under very mild conditions that E G [ | θ | ] < ∞ and the critical point of a Bayes test is finite, δ ˜ n is shown to be asymptotically optimal, and the associated regret converges to zero at a rate O ( n - 1 ( ln n ) 1.5 ) where n is the number of past experiences available when the current component decision problem is considered. This rate achieves the optimal rate which was established by Gupta and Li (Optimal rate of convergence of monotone empirical Bayes tests for a normal mean. Technical Report 01-03, Department of Statistics, Purdue University.) for variance σ 2 known case.