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

Let $X_1,X_2,\dots$ be i.i.d. random variables with common density $f(x-\Theta)$ depending on a location parameter $\Theta\in R^1$. Consider testing the null hypothesis $H_0:\Theta=0$ against $H_1:\Theta\ne0$ and let $\{T_n(X_1,X_2,\dots,X_n)\}$ be a sequence of test statistics. The property of local asymptotic optimality of $\{T_n\}$ in the Bahadur sense means that the exact slope $C_T(\Theta)$ of $\{T_n\}$ is equivalent to
$$ 2K(\Theta)=2\int_{-\infty}^\infty\ln\frac{f(x-\Theta)}{f(x)}f(x-\Theta)\,dx $$
when $\Theta\to0$. The aim of the paper is to obtain characterizations of densities $f$ for which test statistics such as the sample mean Kolmogorov–Smirnov and $\omega^2$ are locally asymptotically optimal. The typical result is as follows: under some conditions $\omega^2$-criterion is locally asymptotically optimal iff $f(x)=(\pi\ch x)^{-1}$, possibly with other location and scale. Similar results are obtained in the two-sample case.