Testing in two parameter exponential distributions via empirical Bayes method

A monotone empirical Bayes test δ n * for testing the hypotheses H 0:θ≤θ 0 versus H 1:θ>θ 0 using a linear error loss in the two-parameter exponential distribution having probability density \(p(x\vert \theta,\beta)=\frac{1}{\beta}\exp(-(x-\theta)/\beta)\) , x>θ>0, with unknown scale parameter β is considered. The asymptotic optimality of δ n * is studied and under certain regularity conditions, it is shown that the regret of δ n * converges to zero at a rate \(n^{-1+\frac{1}{2(r+\alpha)+1}}\) , where n is the number of past data available and r is some related positive integer with 0≤α≤1. The rate improves significantly the result of [10] under unknown β.