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 S T and V T of possible intensities s T and v T , respectively, the minimax testing of the composite hypothesis H 0 : s T ∈ S T against the composite alternative H 1 : v T ∈ V T is investigated. It is assumed that a pair of intensities s T 0 ∈ S T and v T 0 ∈ V T are chosen, and the “Likelihood-Ratio” test for intensities s T 0 and v T 0 is used for testing composite hypotheses H 0 and H 1 . The case, when the 1-st kind error probability α is fixed and we are interested in the minimal possible 2-nd kind error probability β ( S T , V T ) , is considered. What are the maximal sets S ( s T 0 , v T 0 ) and V ( s T 0 , v T 0 ) , which can be replaced by the pair of intensities ( s T 0 , v T 0 ) without essential loss for testing performance ? In the asymptotic case ( T → ∞ ) those maximal sets S ( s T 0 , v T 0 ) and V ( s T 0 , v T 0 ) are described.