EXACT SLOPES OF TEST STATISTICS FOR THE MULTIVARIATE EXPONENTIAL FAMILY

The objective of this paper is to investigate exact slopes of test sta tistics {T-n} when the random vectors X-1, ..., X-n are distributed ac cording to an unknown member of an exponential family {P-theta: theta is an element of Omega}. Here Omega is a parameter set. We will be con cerned with the hypothesis testing problem of H-0: theta is an element of Omega(0) vs H-1: theta is not an element of Omega(0) where Omega(0 ) is a subset of Omega. It will be shown that for an important class o f problems and test statistics the exact slope of {T-n} at eta in Omeg a - Omega(0) is determined by the shortest Kullback-Leibler distance f rom {theta: T-n(lambda(theta)) = T-n(lambda(eta))} to Omega(0), lambda (theta) = E-theta(X).