On the asymptotical effectiveness of testing a simple hypothesis against a composite alternative

Let $X_\varepsilon$ be an observations with a distribution $P_\theta^\varepsilon$, $\theta\in\Theta$, where the parametric space $\Theta$ is an open subset if the real line, $\varepsilon$ is a real parameter, $\varepsilon\to\varepsilon_0$ (for example, $\varepsilon$ is the number of discrete observations in the sample $X_\varepsilon$ or the length of a continuous process realisation $X_\varepsilon$: $\varepsilon_0=\infty$). On the basis of Вayes' approach we consider the problem of testing the hypothesis $H_0$: $\theta=\xi$ against the hypothesis $H_1$: $\theta$ is a random variable having a priori distribution with the density $\pi(\theta)$. If the probability of the error of the second kind is fixed, then the optimal test (which minimizes the probability of an error of the first kind) is based on the likelihood ratio
$$ \frac{dP_{H_1}^\varepsilon}{dP_{H_0}^\varepsilon}= \int_\Theta\frac{dP_\theta^\varepsilon}{dP_\xi^\varepsilon}\pi(\theta)\,d\theta $$
It is shown that the methods elaborated in [1]–[3] enable us to prove the asymptotic optimality of likelihood ratio test and to receive the asymptotically exact estimates for the probability of error of the first kind for the optimal test. We extend also some results of [5] on a class of models considered in [1]–[4].