Nonparametric regression estimation onto a Poisson point process covariate

Cadre, Benoît; Truquet, Lionel

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

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$𝔼$(${r̂\left(X\right)-r\left(X\right))}^2$.

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

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