Unconditional tests of goodness of fit for the intensity of time-truncated nonhomogeneous Poisson processes

Procedures for testing trends in the intensity functions of nonhomogeneous Poisson processes are based mostly on conditioning on the number of failures observed in (0. t] with fixed t. We study an unconditional test based on the time-truncated data that enables meaningful asymptotics as t --> infinity. We show that the asymptotic test is conservative and that its power quickly comes close to the power of the uniformly most powerful unbiased test for the power-law alternatives. Moreover, for the goodness of fit of a specified intensity, the exact test has more power than the test based on the conditional approach. We illustrate the procedure using a real dataset.