Let be a real random variable and be a Poisson point process. We investigate rates of convergence of a nonparametric estimate of the regression function r(x) = (Y|X = x), based on independent copies of the pair . The estimator is constructed using a Wiener–Itô decomposition of . In this infinite-dimensional setting, we first obtain a finite sample bound on the expected squared difference (. Then, under a condition ensuring that the model is genuinely infinite-dimensional, we obtain the exact rate of convergence of ln(.
DOI : 10.1051/ps/2014023
Mots-clés : Regression estimation, Poisson point process, Wiener–Itô decomposition, rates of convergence
@article{PS_2015__19__251_0,
author = {Cadre, Beno{\^\i}t and Truquet, Lionel},
title = {Nonparametric regression estimation onto a {Poisson} point process covariate},
journal = {ESAIM: Probability and Statistics},
pages = {251--267},
publisher = {EDP-Sciences},
volume = {19},
year = {2015},
doi = {10.1051/ps/2014023},
mrnumber = {3412645},
zbl = {1392.62109},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2014023/}
}
TY - JOUR AU - Cadre, Benoît AU - Truquet, Lionel TI - Nonparametric regression estimation onto a Poisson point process covariate JO - ESAIM: Probability and Statistics PY - 2015 SP - 251 EP - 267 VL - 19 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2014023/ DO - 10.1051/ps/2014023 LA - en ID - PS_2015__19__251_0 ER -
%0 Journal Article %A Cadre, Benoît %A Truquet, Lionel %T Nonparametric regression estimation onto a Poisson point process covariate %J ESAIM: Probability and Statistics %D 2015 %P 251-267 %V 19 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2014023/ %R 10.1051/ps/2014023 %G en %F PS_2015__19__251_0
Cadre, Benoît; Truquet, Lionel. Nonparametric regression estimation onto a Poisson point process covariate. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 251-267. doi : 10.1051/ps/2014023. http://www.numdam.org/articles/10.1051/ps/2014023/
D. Applebaum, Universal Malliavin calculus in Fock and Lévy-Itô Spaces. Commun. Stoch. Anal. (2009) 119–141. | MR
and , Remark on the finite-dimensional character of certain results of functional statistics. C.R. Acad. Sci. 351 (2013) 139–141. | MR | Zbl
, and , Rates of convergence of the functional -nearest neighbor estimate. IEEE Trans. Inf. Theory 56 (2010) 2034–2040. | MR | Zbl
G. Biau, B. Cadre and Q. Paris, Cox process functional learning. To appear in Stat. Int. Stoch. Processes (2015). | MR
, and , Supervised classification for a family of Gaussian functional models. Scand. J. Statist. 38 (2011) 480–498. | MR | Zbl
, Supervised classification of diffusion paths. Math. Methods Statist. 22 (2013) 213–225. | MR | Zbl
L. Gyor¨fi, M. Kohler, A. Krzyżak and H. Walk, A distribution-Free Theory of Nonparametric Regression. Springer-Verlag, New-York (2002). | MR | Zbl
, Spectral type of the shift transformation of differential processes with stationary increments. Trans. Amer. Math. Soc. 81 (1956) 253–263. | MR | Zbl
J.F.C. Kingman, Poisson Processes. In Oxf. Stud. Probab. Oxford Science publications, 1st ed. (1993). | MR | Zbl
and , Poisson process Fock space representation, chaos expansion and covariance inequalities. Probab. Theory Relat. Fields 150 (2011) 663–690. | MR | Zbl
, Stationaire zufällige Mae auf lokalkompakten abelschen Gruppen. Z. Wahrsch. verw. Geb. 9 (1967) 36–58. | MR | Zbl
D. Nualart and J. Vives, Anticipative calculus for the Poisson process based on the Fock space. Séminaire de Probabilités XXIV. Lect. Notes Math. (1990) 154–165. | MR | Zbl
J.O. Ramsay and B.W. Silverman, Functional Data Analysis. Springer-Verlag, New-York (1997). | MR | Zbl
B.W. Silverman, Density Estimation for Statistics and Data Analysis. Springer-Verlag, New-York (1986). | MR | Zbl
, The homogeneous chaos. Am. J. Math. 60 (1938) 897–936. | JFM | MR
Cité par Sources :
