Abstract
The objective of this paper is to investigate exact slopes of test statistics {Tn} when the random vectors X1, ..., Xn are distributed according to an unknown member of an exponential family {Pθ; θ∈Ω. Here Ω is a parameter set. We will be concerned with the hypothesis testing problem of H0θ∈Ω0 vs H1: 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 = Eθ)(X).
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