Abstract: | The objective of this paper is to investigate exact slopes of test statistics { Tn } when the random vectors X 1, ..., Xn are distributed according to an unknown member of an exponential family { P θ; θ∈Ω. Here Ω is a parameter set. We will be concerned with the hypothesis testing problem of H 0θ∈Ω0 vs H 1: θ∉Ω0 where Ω0 is a subset of Ω. It will be shown that for an important class of problems and test statistics the exact slope of { Tn } at η in Ω−Ω0 is determined by the shortest Kullback–Leibler distance from {θ: Tn (λ(θ)) = Tn (λ(π))} to Ω0, λθ = E θ)( X ). |