Optimal sequences of tests for several polynomial schemes of trials

Sequences $\{\omega^n\}$ of tests $\omega^n$ are considered for solution of the problem of choice of the true polynomial scheme of trials by using frequencies for $n$ independent trials made according to one of $m$ possible schemes with the same set of outcomes. Let $\alpha_s(\omega^n )$ be the probability not to accept the true $s$th scheme, $s=1,\dots,m$. The behavior of the quantity $\max_{s\in J}\alpha_s (\omega^n)$ is studied for given $J\subseteq \{1,\dots,m\}$ and $n \to\infty$ for sequences $\{\omega^n\}$ from the set $N$, characterized by the property that the probabilities $\alpha_t(\omega^n)$ for $t\in I$, $I\subseteq\{1,\dots,m\}$, satisfy certain conditions, for example, $\alpha_t(\omega^n )\le\alpha_t < 1$ or $\alpha_t(\omega^n )\le a_t\exp(-nv_t)$ for all $n\ge n_0$. The sequences $\{g^n\}\in N$ are given and the quantity $M(N,J)\ge 0$ is computed such that $\max_{s\in J}\alpha_s(g^n)=\exp(-nM(N,J)+o(n))$ and there is no sequence $\{\omega^n\} \in N$, for which $\max_{s\in J}\alpha_s(\omega^n)=\exp(-nM+o(n))$, $M>M(N,J)$. The upper bounds for $\alpha_t(g^n)$, $t=1,\dots,m$, tending to 0 as $n\to\infty$ are explicitly computed.