Hypothesis testing for nonhomogeneous poisson processes with singular alternatives

Abstract

We study hypothesis testing for nonhomogeneous Poisson processes under singular alternatives, where the null intensity may vanish while the alternative remains positive. We develop a unified framework combining exact methods, Edgeworth expansions, and large deviation principles. We derive the Neyman--Pearson optimal test accounting for singular components, establish a central limit theorem for the log-likelihood ratio, and obtain a first-order Edgeworth expansion yielding refined critical values. A large deviation analysis characterizes the exponential decay of the Type II error. Numerical simulations confirm improved size control and power. This framework provides a unified treatment of testing in non-regular Poisson models.