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

Ya. Yu. Nikitin

Zapiski Nauchnykh Seminarov POMI, Studies in mathematical statistics. Part V, Tome 108 (1981), pp. 119-133

Résumé

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.