Nonparametric tests for Cox processes

In a functional setting, we elaborate and study two test statistics to highlight the Poisson nature of a Cox process when n copies of the process are available. Our approach involves a comparison of the empirical mean and the empirical variance of the functional data and can be seen as an extended version of a classical overdispersion test for count data. The limiting distributions of our statistics are derived using a functional central limit theorem for càdlàg martingales. Our procedures are easily implementable and do not require any knowledge on the covariate. We address a theoretical comparison of the asymptotic power of our tests under some local alternatives. A numerical study reveals the good performances of the method. We also present two applications of our tests to real datasets.