Local asymptotic efficiency of a sequential probability ratio testfor $d$-guarantee discriminationof composite hypotheses

A sequential Wald test for discrimination of two simple hypotheses $\theta=\theta_1$ and $\theta=\theta_2$ with boundaries $A$ and $B$ is applied to distinguish composite hypotheses $\theta<\theta_0$ and $\theta>\theta_0$, the parameters $\theta_1, \theta_2, A$, and $B$ being chosen in such a way that $d$-posteriori probabilities of errors do not exceed the given restrictions $\beta_0$ and $\beta_1$. An asymptotic behavior of boundaries $A, B$ and the average observation time are studied when $\beta=\max\{\beta_0, \beta_1\}\to 0$. An asymptotic $(\beta\to 0)$ comparison is made of ${\mathbb{E}}_{\theta}\nu$ with the least given number of observations necessary for discrimination of composite hypotheses with the same restrictions $\beta_0, \beta_1$ on $d$-posteriori probabilities of errors. It is shown that the minimum (in a neighborhood of the point $\theta=\theta_0$) gain of the average observation time makes up 25%. Therefore, there are sequential tests within the bounds of a $d$-posteriori approach that give a gain in the size of observations for every value of a parameter tested.