Characterization of distributions by the local asymptotic optimality property of tests statistics

Abstract

Let X1, X2,... be a sequence of independent, identically distributed random variables with density f(x−θ), θ ε R1. We consider the problem of testing the hypothesis H0∶=0 against H1∶ ≠ 0 on the basis of a sequence of test statistics {Tn(X1,...,Xn). In accordance with the Bahadur theory, a measure of the asymptotic efficiency of {Tn} is its exact slope Ct(θ). One says that {Tn} is locally asymptotically optimal in the Bahadur sense if Ct(θ) ip 2K(θ), θ → 0, where . The purpose of the paper is to characterize densities f for which the property of local asymptotic optimality is shared by such commonly used statistics as the sample mean, the Kolmogorov-Smirnov statistic, the sign statistic, Ω2, etc. Under certain restrictions on f one proves, for example, that the sequence of statistics u is locally asymptotically optimal only for the “hyperbolic cosine” distribution, while the Kolmogorov statistic only for the Laplace distribution. At the end of the paper one obtains similar results for the two-sample case, in particular, for a large class of linear rank statistics.