The structure of the class of absolutely admissible tests

Let $Q$ be a distribution in $R^k$ which is absolutely continuous with respect to the Lebesgue measure, and let $Q_\lambda$, $\lambda\in\Lambda\subseteq R^k$ be an exponential family such that
$$ dQ_\lambda/dQ=b(\lambda)\exp\{(\lambda,y)\},\qquad y\in R^k, $$
where $(y,\lambda)$ denotes the scalar product in $R^k$ and $B(\lambda)$ is a norming constant. Let $y$ be an observation of the random variable $Y$ with distribution $Q_\lambda$. Let $\Phi_\varepsilon$ be a complete class of admissible tests in the problem of testing the hypothesis $H_0\colon\lambda=0$ against the alternatives $H_\varepsilon$: $\lambda\ne 0$, $|\lambda|\le\varepsilon$, and $\Phi_0=\bigcap\limits_{\varepsilon>0}\Phi_\varepsilon$. It is proved that the class $\Phi_0$ consists of tests the acceptance regions of which are either the ellipsoidal cylinder or the half-space. Moreover, it is shown that the necessary condition for the test $\varphi$ to belong to the class $\Phi_R$ for any $R>0$ is the following one: the boundary of the acceptance region of $\varphi$ is an analytic $(k-1)$-dimensional real manifold in $R^k$. In particular, the likelihood ratio test for normal distribution $N(\lambda,I)$ and alternatives $0<|\lambda|\le R$, $\lambda_1\ge 0$ is unadmissible.