Simultaneous Testing of Change-Point Location and of a Regular Parameter by Poisson Observations

The problem of hypothesis testing is considered in the case of observation of an inhomogeneous Poisson process with an intensity function depending on two parameters. It is supposed that the dependence on the first of them is sufficiently regular, while the second one is a change-point location. Under the null hypothesis the parameters take some known values, while under the alternative they are greater (with at least one of the inequalities being strict). Four test are studied: the general likelihood ratio test (GLRT), the Wald's test and two Bayesian tests (BT1 and BT2). For each of the tests, expressions allowing to approximate its threshold and its limit power function by Monte Carlo numerical simulations are derived. Moreover, for the GLRT, an analytic equation for the threshold and an analytic expression of the limit power function are obtained. Finally, numerical simulations are carried out and the performance of the tests is discussed.