Nonparametric regression estimation onto a Poisson point process covariate

International audienceLet Y be a real random variable and X be a Poisson point process. We investigate rates of convergence of a nonparametric estimatê r(x) of the regression function $r(x) = E(Y |X = x)$, based on n independent copies of the pair (X,Y). The estimatorˆrestimatorˆ estimatorˆr is constructed using a Wiener-Itô decomposition of $r(X)$. In this infinite-dimensional setting, we first obtain a finite sample bound on the expected squared difference $E(ˆ r(X) − r(X)) 2$. Then, under a condition ensuring that the model is genuinely infinite-dimensional, we obtain the exact rate of convergence of ln $E(ˆ r(X) − r(X)) 2$