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On pointwise adaptive curve estimation basedon inhomogeneous data

Published online by Cambridge University Press:  17 August 2007

Stéphane Gaïffas
Stéphane Gaïffas*
Affiliation:
Laboratoire de Probabilités et Modèles Aléatoires, U.M.R. CNRS 7599 and Université Paris 7, 175 rue du Chevaleret, 75013 Paris, France; gaiffas@math.jussieu.fr
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Abstract

We want to recover a signal based on noisy inhomogeneous data (the amount of data can vary strongly on the estimation domain). We model the data using nonparametric regression with random design, and we focus on the estimation of the regression at a fixed point x 0 with little, or much data. We propose a method which adapts both to the local amount of data (the design density is unknown) and to the local smoothness of the regression function. The procedure consists of a local polynomial estimator with a Lepski type data-driven bandwidth selector, see for instance Lepski et al. [Ann. Statist.25 (1997) 929–947]. We assess this procedure in the minimax setup, over a class of function with local smoothness s >0 of Hölder type. We quantify the amount of data at x 0 in terms of a local property on the design density called regular variation, which allows situations with strong variations in the concentration of the observations. Moreover, the optimality of the procedure is proved within this framework.

Keywords

Adaptive estimationinhomogeneous datanonparametric regressionrandom design.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 11: Special Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthdaySpecial Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthday , June 2007 , pp. 344 - 364
Copyright
© EDP Sciences, SMAI, 2007

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References

Antoniadis, A., Gregoire, G. and Vial, P., Random design wavelet curve smoothing. Statist. Probab. Lett. 35 (1997) 225–232. CrossRef
Baraud, Y., Model selection for regression on a random design. ESAIM Probab. Statist. 6 (2002) 127–146 (electronic). CrossRef
N.H. Bingham, C.M. Goldie and J.L. Teugels, Regular Variation. Encyclopedia of Mathematics and its Applications, Cambridge University Press (1989).
Brown, L. and Cai, T., Wavelet shrinkage for nonequispaced samples. Ann. Statist. 26 (1998) 1783–1799.
Brown, L.D. and Low, M.G., A constrained risk inequality with applications to nonparametric functional estimations. Ann. Statist. 24 (1996) 2524–2535.
T.T. Cai, M. Low and L.H. Zhao, Tradeoffs between global and local risks in nonparametric function estimation. Tech. rep., Wharton, University of Pennsylvania, http://stat.wharton.upenn.edu/~tcai/paper/html/Tradeoff.html (2004).
Delouille, V., Simoens, J. and Von Sachs, R., Smooth design-adapted wavelets for nonparametric stochastic regression. J. Amer. Statist. Soc. 99 (2004) 643–658. CrossRef
Fan, J. and Gijbels, I., Data-driven bandwidth selection in local polynomial fitting: variable bandwidth and spatial adaptation. J. Roy. Statist. Soc. Ser. B. Methodological 57 (1995) 371–394.
J. Fan and I. Gijbels, Local polynomial modelling and its applications. Monographs on Statistics and Applied Probability, Chapman & Hall, London (1996).
Gaïffas, S., Convergence rates for pointwise curve estimation with a degenerate design. Mathematical Methods of Statistics 1 (2005) 1–27. Available at http://hal.ccsd.cnrs.fr/ccsd-00003086/en/
Goldenshluger, A. and Nemirovski, A., On spatially adaptive estimation of nonparametric regression. Math. Methods Statist. 6 (1997) 135–170.
G. Kerkyacharian and D. Picard, Regression in random design and warped wavelets. Bernoulli, 10 (2004) 1053–1105.
Lepski, O.V., Asymptotically minimax adaptive estimation i: Upper bounds, optimally adaptive estimates. Theory Probab. Applic. 36 (1988) 682–697. CrossRef
O.V. Lepski, On a problem of adaptive estimation in Gaussian white noise. Theory Probab. Appl., 35 (1990) 454–466.
Lepski, O.V., Mammen, E. and Spokoiny, V.G., Optimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimates with variable bandwidth selectors. Ann. Statist. 25 (1997) 929–947.
Lepski, O.V. and Spokoiny, V.G., Optimal pointwise adaptive methods in nonparametric estimation. Ann. Statist. 25 (1997) 2512–2546.
V. Maxim, Restauration de signaux bruités sur des plans d'experience aléatoires. Ph.D. thesis, Université Joseph Fourier, Grenoble 1 (2003).
Spokoiny, V.G., Estimation of a function with discontinuities via local polynomial fit with an adaptive window choice. Ann. Statist. 26 (1998) 1356–1378.
Stone, C.J., Optimal rates of convergence for nonparametric estimators. Ann. Statist. 8 (1980) 1348–1360. CrossRef
A. Tsybakov, Introduction à l'estimation non-paramétrique. Springer (2003).
Wong, M.-Y. and Zheng, Z., Wavelet threshold estimation of a regression function with random design. J. Multivariate Anal. 80 (2002) 256–284.