On Neyman–Pearson minimax detection of Poisson process intensity

The problem of the minimax testing of the Poisson process intensity $${\mathbf{s}}$$ s is considered. For a given intensity $${\mathbf{p}}$$ p and a set $$\mathcal{Q}$$ Q , the minimax testing of the simple hypothesis $$H_{0}: {\mathbf{s}} = {\mathbf{p}}$$ H 0 : s = p against the composite alternative $$H_{1}: {\mathbf{s}} = {\mathbf{q}},\,{\mathbf{q}} \in \mathcal{Q}$$ H 1 : s = q , q ∈ Q is investigated. The case, when the 1-st kind error probability $$\alpha $$ α is fixed and we are interested in the minimal possible 2-nd kind error probability $$\beta ({\mathbf{p}},\mathcal{Q})$$ β ( p , Q ) , is considered. What is the maximal set $$\mathcal{Q}$$ Q , which can be replaced by an intensity $${\mathbf{q}} \in \mathcal{Q}$$ q ∈ Q without any loss of testing performance? In the asymptotic case ( $$T\rightarrow \infty $$ T → ∞ ) that maximal set $$\mathcal{Q}$$ Q is described.