Nonparametric regression estimation onto a Poisson point process covariate

Let Y be a real random variable and X be a Poisson point process. We investigate rates of convergence of a nonparametric estimate r̂(x) of the regression function r(x) = \hbox{$\mathbb E$}(Y|X = x), based on n independent copies of the pair (X,Y). The estimator r̂ 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 \hbox{$\mathbb E$}(r̂(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\hbox{$\mathbb E$}(r̂(X) - r(X))2.