Nonparametric regression estimation onto a Poisson point process covariate^∗

¹ IRMAR, ENS Rennes, CNRS, UEB, Campus de Ker Lann, Avenue Robert Schuman, 35170 Bruz, France
cadre@ens-rennes.fr
² IRMAR, Ensai, CNRS, UEB, Campus de Ker Lann, Avenue Robert Schuman, 35170 Bruz, France
truquet@ensai.fr

Received: 13 March 2014
Revised: 18 September 2014

Abstract

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) = (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 (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(r̂(X) - r(X))².