Local Bahadur optimality and characterization problems

Let $X_1,X_2,\dots$ be i. i. d. observations with a common distribution function $G(x;\theta)$. Consider the problem of testing the null hypothesis $H_0:\,\theta=0$ against $H_1:\,\theta>0$ on the basis of a sequence of test statistics $\{T_n=T_n(X_1,\dots,X_n)\}$ with an exact Bahadur slope $c_T(\theta)$. The sequence $\{T_n\}$ is said to be locally optimal if $c_T(\theta)\sim 2K(\theta)$, $\theta\to 0$, where $K(\theta)$ is the Kullback–Leibler information number. The aim of the paper is to describe the class of distribution functions $G(x,\theta)$ (the domain of local Bahadur optimality) for which some well-known nonparametric statistics such as Kolmogorov–Smirnov $\omega^2$, their two-sample analogues and linear rank statistics are locally optimal. If $\theta$ is a location or a scale parameter, this domain consists of a single law, e. g. of the Laplace distribution for Kolmogorov–Smirnov statistic and the hyperbolic cosine distribution for $\omega^2$-statistic in the location case.