Consider an inhomogeneous Poisson process X on [0, T] whose unk-nown intensity function “switches” from a lower function g* to an upper function h* at some unknown point ϑ* that has to be identified. We consider two known continuous functions g and h such that g*(t) ⩽ g(t) < h(t) ⩽ h*(t) for 0 ⩽ t ⩽ T. We describe the behavior of the generalized likelihood ratio and Wald’s tests constructed on the basis of a misspecified model in the asymptotics of large samples. The power functions are studied under local alternatives and compared numerically with help of simulations. We also show the following robustness result: the Type I error rate is preserved even though a misspecified model is used to construct tests.