Analogue of the Neyman–Pearson lemma for several simple hypotheses

We consider the following problem on testing $r$ simple hypotheses: in the set $K^{\alpha}$ of tests with weighed sum of errors of the $i$th kind, $1\le i \le k$, at most $\alpha$, it is required to single out a subset $\Pi^{\mathrm{opt}}$ of tests at which the minimum of the weighed sum of errors of the $i$th kind, $k < i \le r$, is attained. We show that the set $\Pi^{\mathrm{opt}}$ is the intersection of the set $K^{\alpha}$ (or, depending on $\alpha$, of the boundary of $K^{\alpha}$) with some auxiliary set of Bayesian tests. An algorithm for construction of optimal tests is given. The main theorem of the paper generalizes the well-known Neyman–Pearson lemma.