Exact Slopes of Test Statistics for the Multivariate Exponential Family

The objective of this paper is to investigate exact slopes of test statistics { T n } when the random vectors X 1 , ..., X n 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 0 θ∈Ω 0 vs H 1 : θ∉Ω 0 where Ω 0 is a subset of Ω. It will be shown that for an important class of problems and test statistics the exact slope of { T n } at η in Ω−Ω 0 is determined by the shortest Kullback–Leibler distance from {θ: T n (λ(θ)) = T n (λ(π))} to Ω 0 , λ θ = E θ )( X ).