Strong large deviations for Rao test score and GLRT in exponential families

Exact asymptotics are derived for composite hypothesis testing between two product probability measures Pⁿ vs Qⁿ, subject to a type-I error-probability constraint ε. Here P is known but Q is an unknown element of a given d-dimensional regular exponential family. We study the Rao score test, which is a quadratic approximation to the GLRT. The type-II error probability is shown to vanish as equation where D and V are respectively the Kullback-Leibler divergence and the variance of information divergence between P and Q; τ(ε; d) is the 1 - ε quantile for the χ_d² distribution; and the constants β_d > 0 and γ_d are explicitly identified. The asymptotic regret relative to the Neyman-Pearson test (which knows Q) is reflected in the coefficient τ(ε; d), as is the cost of dimensionality. Looser asymptotics (with O(1) in place of ε_d) are obtained for the GLRT.