We propose several results concerning the construction of Kolmogorov-Smirnov and Cramér-von Mises type goodness-of -fit tests for a two-dimensional inhomogeneous Poisson process.  Under the basic hypothesis, the observed process is a bivariate inhomogeneous Poisson process with discontinuous intensity function. For this model, we propose tests which are asymptotically distribution free and consistent.

composite hypotheses testing, Poisson process, goodness-of-fit test, parameter estimation, singular estimation, distribution free test.