Nonparametric regression estimation onto a Poisson point process covariate

Cadre, Benoît¹ ; Truquet, Lionel² 

¹ IRMAR, ENS Rennes, CNRS, UEB, Campus de Ker Lann, Avenue Robert Schuman, 35170 Bruz, France
² IRMAR, Ensai, CNRS, UEB, Campus de Ker Lann, Avenue Robert Schuman, 35170 Bruz, France

ESAIM: Probability and Statistics, Tome 19 (2015), pp. 251-267

Résumé

Let $Y$ be a real random variable and $X$ be a Poisson point process. We investigate rates of convergence of a nonparametric estimate $r̂\left(x\right)$ of the regression function r(x) = $𝔼$(Y|X = x), based on $n$ independent copies of the pair $(X,Y)$. The estimator $r̂$ is constructed using a Wiener–Itô decomposition of $r\left(X\right)$. In this infinite-dimensional setting, we first obtain a finite sample bound on the expected squared difference $𝔼$(${r̂\left(X\right)-r\left(X\right))}^2$. Then, under a condition ensuring that the model is genuinely infinite-dimensional, we obtain the exact rate of convergence of ln$𝔼$(${r̂\left(X\right)-r\left(X\right))}^2$.

Détail

Reçu le : 2014-03-13
DOI : 10.1051/ps/2014023

Classification : 62G05, 62G08
Keywords: Regression estimation, Poisson point process, Wiener–Itô decomposition, rates of convergence