Strong large deviations for composite hypothesis testing

A simple hypothesis P is tested against a composite hypothesis ℚ^j, j ∈ {1,2, ...,k}, each ℚ^j being a product of n probability distributions. We consider the set of achievable false-positive error probability vectors for a generalized Neyman-Pearson test under a constraint on the probability of correct detection under P. Exact asymptotics (as n → ∞) are derived for this set, in particular the set is determined within an O(1) term.