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

Let X₁, X₂,... be a sequence of independent, identically distributed random variables with density f(x–), R¹. We consider the problem of testing the hypothesis H₀=0 against H₁ 0 on the basis of a sequence of test statistics {T_n(X₁,...,X_n). In accordance with the Bahadur theory, a measure of the asymptotic efficiency of {T_n} is its exact slope C_t(). One says that {T_n} is locally asymptotically optimal in the Bahadur sense if C_t() ip 2K(), 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, ², 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.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 108, pp. 119–133, 1981.