On hypothesis tests in misspecified change-point problems for a Poisson process.

Consider an inhomogeneous Poisson processXon [0,T] whose unk-nown intensity function “switches” from a lower functiong to an upper functionh at some unknown point ϑ that has to be identified. We consider two known continuous functionsgandhsuch thatg (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. [ABSTRACT FROM AUTHOR]