Abstract. For a Poisson point process X, Itô's famous chaos expansion implies that every square integrable regression function r with covariate X can be decomposed as a sum of multiple stochastic integrals called chaos. In this paper, we consider the case where r can be decomposed as a sum of δ chaos. In the spirit of Cadre and Truquet [ESAIM: PS 19 (2015) 251-267], we introduce a semiparametric estimate of r based on i.i.d. copies of the data. We investigate the asymptotic minimax properties of our estimator when δ is known. We also propose an adaptive procedure when δ is unknown.

International audienceIn 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 c`adì ag 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 data sets