On minimax robust testing of composite hypotheses on Poisson process intensity

The problem on the minimax testing of a Poisson process intensity is considered. For a given disjoint sets $${{\mathcal {S}}}_T$$ S T and $${{\mathcal {V}}}_T$$ V T of possible intensities $${{\mathbf {s}}}_{T}$$ s T and $${{\mathbf {v}}}_{T}$$ v T , respectively, the minimax testing of the composite hypothesis $$H_{0}: {{\mathbf {s}}_T} \in {{\mathcal {S}}}_T$$ H 0 : s T ∈ S T against the composite alternative $$H_{1}: {{\mathbf {v}}_T} \in {{\mathcal {V}}}_T$$ H 1 : v T ∈ V T is investigated. It is assumed that a pair of intensities $${{\mathbf {s}}_T^{0}} \in {{\mathcal {S}}}_T$$ s T 0 ∈ S T and $${{\mathbf {v}}_T^{0}} \in {{\mathcal {V}}}_T$$ v T 0 ∈ V T are chosen, and the “Likelihood-Ratio” test for intensities $${{\mathbf {s}}_T^{0}}$$ s T 0 and $${{\mathbf {v}}_T^{0}}$$ v T 0 is used for testing composite hypotheses $$H_{0}$$ H 0 and $$H_{1}$$ H 1 . 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 $$\beta ({{\mathcal {S}}}_T,{{\mathcal {V}}}_T)$$ β ( S T , V T ) , is considered. What are the maximal sets $${{\mathcal {S}}}({{\mathbf {s}}}_{T}^{0},{{\mathbf {v}}}_{T}^{0})$$ S ( s T 0 , v T 0 ) and $${{\mathcal {V}}}({{\mathbf {s}}}_{T}^{0},{{\mathbf {v}}}_{T}^{0})$$ V ( s T 0 , v T 0 ) , which can be replaced by the pair of intensities $$({{\mathbf {s}}_T^{0}},{{\mathbf {v}}_T^{0}})$$ ( s T 0 , v T 0 ) without essential loss for testing performance ? In the asymptotic case ( $$T\rightarrow \infty $$ T → ∞ ) those maximal sets $${{\mathcal {S}}}({{\mathbf {s}}}_{T}^{0},{{\mathbf {v}}}_{T}^{0})$$ S ( s T 0 , v T 0 ) and $${{\mathcal {V}}}({{\mathbf {s}}}_{T}^{0},{{\mathbf {v}}}_{T}^{0})$$ V ( s T 0 , v T 0 ) are described.