On Neyman–Pearson minimax detection of Poisson process intensity

Abstract

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