Asymptotics of the minimum sufficient number of observations for $d$-guaranteed discrimination of two-sided hypotheses

We consider the problem of constructing guarantee procedures of statistical inference with fixed minimal observation number $n^*$ for discrimination of two hypotheses $H_0\colon\theta\in[\theta_1,\theta_2]$ and $H_1\colon\theta\notin[\theta_1,\theta_2]$ with a one-dimensional parameter $\theta$ under the so-called $d$-posterior approach. Here, constraints are placed on the conditional probabilities for the validity of one or another hypothesis under the condition that this hypothesis is rejected. We give an asymptotic formula for $n^*$ in a scheme with severe (tending to zero) constraints on these conditional probabilities of hypotheses. Earlier, Volodin and Novikov found a similar formula for discrimination of one-sided hypotheses. In the present paper, the proof of the asymptotic formula is carried out under weaker constraints on the probability model. The accuracy of our formula is illustrated numerically for some probability models.