On Neyman–Pearson minimax detection of Poisson process intensity

The problem of the minimax testing of the Poisson process intensity s is considered. For a given intensity p and a set Q , the minimax testing of the simple hypothesis H 0 : s = p against the composite alternative H 1 : s = q , q ∈ Q is investigated. 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 ( p , Q ) , is considered. What is the maximal set Q , which can be replaced by an intensity q ∈ Q without any loss of testing performance? In the asymptotic case ( T → ∞ ) that maximal set Q is described.