A Theorem on Admissible Tests and Its Application to an Asymptotical Problem of Testing Hypotheses

Let $X_1,…,X_n$ be independent observations of a real random variable having a density $f(x,\vartheta)$, $\vartheta$ being a parameter from $s$-dimensional Euclidean space $E^s$. The hypothesis $\vartheta=0$ is tested. A test is said to be asymptotically (as $n\to\infty$) optimal if it has asymptotically best average power with respect to a given family of probability measures on the space of values of normed parameter $\theta=\vartheta/\sqrt n$ (see (2.1), (2.2)). It is shown that the problem, of constructing an asymtotically optimal test can be reduced to that of constructing an optimal (in the corresponding sense) test for some family of normal distributions in $E^s$ that differ only in locations.
In the proof the following result is used. Let $Q_0$ be a distribution in $E^s$ and $\{Q_0\}$, $\theta\in\Theta\subset E^s$ be the exponential family such that $dQ_\theta/dQ_0=\mathbf C(\theta)\exp(\theta,y)$, $y\in E^s$, $(\theta,y)$ denoting the scalar product. The hypothesis $\theta=0$ is tested. Then the following condition is necessary for a test $\varphi$ to be admissible: there exists a closed convex set $C\subset E^s$ such that (up to a set of $Q_0$-measure zero) $\varphi(y)=1$ if $y\in E^s\setminus C$ and $\varphi(y)=0$ if $y\in C^0$ where $C^0$ is the set of inner points of $C$.