Minimax regression estimation for Poisson coprocess

Cadre, Benoît; Klutchnikoff, Nicolas; Massiot, Gaspar

doi:10.1051/ps/2017004

Cadre, Benoît ¹ ; Klutchnikoff, Nicolas ² ; Massiot, Gaspar ²

¹ IRMAR, ENS Rennes, CNRS, Campus de Ker Lann, Avenue Robert Schuman, 35170 Bruz, France.
² IRMAR, Université de Rennes 2, CNRS, Campus Villejean, Place du recteur Henri le Moal, CS 24307, 35043 Rennes cedex, France.

ESAIM: Probability and Statistics, Tome 21 (2017), pp. 138-158.

Résumé

For a Poisson point process $X$ , Itô’s famous chaos expansion implies that every square integrable regression function $r$ with covariate $X$ can be decomposed as a sum of multiple stochastic integrals called chaos. In this paper, we consider the case where $r$ can be decomposed as a sum of $δ$ chaos. In the spirit of Cadre and Truquet [ESAIM: PS 19 (2015) 251–267], we introduce a semiparametric estimate of $r$ based on i.i.d. copies of the data. We investigate the asymptotic minimax properties of our estimator when $δ$ is known. We also propose an adaptive procedure when $δ$ is unknown.

Reçu le : 2016-11-29
Accepté le : 2017-03-05

MR Zbl

DOI : 10.1051/ps/2017004

Classification : 62G08, 62H12, 62M30
Mots-clés : Functional statistic, poisson point process, regression estimate, minimax estimation

Affiliations des auteurs :

Cadre, Benoît ¹ ; Klutchnikoff, Nicolas ² ; Massiot, Gaspar ²

@article{PS_2017__21__138_0,
     author = {Cadre, Beno{\^\i}t and Klutchnikoff, Nicolas and Massiot, Gaspar},
     title = {Minimax regression estimation for {Poisson} coprocess},
     journal = {ESAIM: Probability and Statistics},
     pages = {138--158},
     publisher = {EDP-Sciences},
     volume = {21},
     year = {2017},
     doi = {10.1051/ps/2017004},
     mrnumber = {3716123},
     zbl = {1395.62086},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps/2017004/}
}

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Cadre, Benoît; Klutchnikoff, Nicolas; Massiot, Gaspar. Minimax regression estimation for Poisson coprocess. ESAIM: Probability and Statistics, Tome 21 (2017), pp. 138-158. doi : 10.1051/ps/2017004. http://www.numdam.org/articles/10.1051/ps/2017004/

Bibliographie
Cité par

G. Biau, F. Cérou and A. Guyader, Rates of convergence of the functional $k$ -nearest neighbor estimate. IEEE Trans. Inform. Theory 56 (2010) 2034–2040. | DOI | MR | Zbl

L. Birgé, Approximation dans les espaces métriques et théorie de l’estimation. L. Z. Wahrscheinlichkeitstheorie verw Gebiete 65 (1983) 181–237. | DOI | MR | Zbl

B. Cadre and L. Truquet, Nonparametric regression estimation onto a Poisson point process covariate. ESAIM: PS 19 (2015) 251–267. | DOI | Numdam | MR | Zbl

G. Chagny and A. Roche, Adaptive estimation in the functional nonparametric regression model. J. Multivariate Anal. 146 (2016) 105–118. | DOI | MR | Zbl

L. Györfi, M. Kohler, A. Krzyzak and H. Walk, A distribution-free theory of nonparametric regression. Springer Science and Business Media (2002). | MR | Zbl

L. Horváth and P. Kokoszka, Inference for functional data with applications. Vol. 200. Springer Science and Business Media (2012). | MR | Zbl

R. Ibragimov and Sh. Sharakhmetov, The exact constant in the Rosenthal Inequality for random variables with mean zero. Theory Probab. Appl. 46 (1999) 127–131. | DOI | MR | Zbl

K. Itô, Spectral type of the shift transformation of differential processes with stationary increments. Trans. Amer. Math. Soc. (1956) 253–263. | MR | Zbl

M. Kohler, A. Krzyżak and H. Walk, Optimal global rates of convergence for nonparametric regression with unbounded data. J. Statist. Plann. Inference 139 (2009) 1286–1296. | DOI | MR | Zbl

G. Last and M.D. Penrose, Poisson process Fock space representation, chaos expansion and covariance inequalities. Probab. Theory Relat. Fields 150 (2011) 663–690. | DOI | MR | Zbl

A. Mas, Lower bound in regression for functional data by representation of small ball probabilities. Electr. J. Statist. 6 (2012) 1745–1778. | MR | Zbl

D. Nualart and J. Vives, Anticipative calculus for the Poisson process based on the Fock space, Séminaire de Probabilitées XXIV. In: Lect. Notes Math. (1990) 154–165. | Numdam | MR | Zbl

J.O. Ramsay and B.W. Silverman, Functional data analysis. Springer Science & Business Media (2005). | MR | Zbl

A.B. Tsybakov, Introduction to nonparametric estimation. Revised and extended from the 2004 French original. Translated by Vladimir Zaiats. Springer Series in Statistics. Springer, New York (2009). | MR | Zbl

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