Electronic Journal of Statistics

Adaptive wavelet multivariate regression with errors in variables

Michaël Chichignoud, Van Ha Hoang, Thanh Mai Pham Ngoc, and Vincent Rivoirard

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In the multidimensional setting, we consider the errors-in- variables model. We aim at estimating the unknown nonparametric multivariate regression function with errors in the covariates. We devise an adaptive estimators based on projection kernels on wavelets and a deconvolution operator. We propose an automatic and fully data driven procedure to select the wavelet level resolution. We obtain an oracle inequality and optimal rates of convergence over anisotropic Hölder classes. Our theoretical results are illustrated by some simulations.

Article information

Electron. J. Statist., Volume 11, Number 1 (2017), 682-724.

Received: January 2016
First available in Project Euclid: 8 March 2017

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Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression

Adaptive wavelet estimator anisotropic regression deconvolution measurement errors

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Chichignoud, Michaël; Hoang, Van Ha; Pham Ngoc, Thanh Mai; Rivoirard, Vincent. Adaptive wavelet multivariate regression with errors in variables. Electron. J. Statist. 11 (2017), no. 1, 682--724. doi:10.1214/17-EJS1238. https://projecteuclid.org/euclid.ejs/1488964114

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  • [1] Bertin, K., Lacour, C. and Rivoirard, V. (2016). Adaptive pointwise estimation of conditional density function., Les Annales de l’IHP 2 939-980.
  • [2] Carroll, R. J., Delaigle, A. and Hall, P. (2009). Nonparametric prediction in measurement error models., J. Amer. Statist. Assoc. 104 993–1003.
  • [3] Chesneau, C. (2010). On adaptive wavelet estimation of the regression function and its derivatives in an errors-in-variables model., Curr. Dev. Theory Appl. Wavelets 4 185–208.
  • [4] Comte, F. and Lacour, C. (2013). Anisotropic adaptive kernel deconvolution., Ann. Inst. Henri Poincaré Probab. Stat. 49 569–609.
  • [5] Comte, F. and Taupin, M. L. (2007). Adaptive estimation in a nonparametric regression model with errors-in-variables., Statist. Sinica 17 1065–1090.
  • [6] Delaigle, A., Hall, P. and Jamshidi, F. (2015). Confidence bands in non-parametric errors-in-variables regression., J. R. Stat. Soc. Ser. B. Stat. Methodol. 77 149–169.
  • [7] Doumic, M., Hoffmann, M., Reynaud-Bouret, P. and Rivoirard, V. (2012). Nonparametric estimation of the division rate of a size-structured population., SIAM J. Numer. Anal. 50 925–950.
  • [8] Du, L., Zou, C. and Wang, Z. (2011). Nonparametric regression function estimation for errors-in-variables models with validation data., Statist. Sinica 21 1093–1113.
  • [9] Fan, J. and Koo, J.-Y. (2002). Wavelet deconvolution., IEEE Trans. Inform. Theory 48 734–747.
  • [10] Fan, J. and Masry, E. (1992). Multivariate regression estimation with errors-in-variables: asymptotic normality for mixing processes., J. Multivariate Anal. 43 237–271.
  • [11] Fan, J. and Truong, Y. K. (1993). Nonparametric regression with errors in variables., Ann. Statist. 21 1900–1925.
  • [12] Gach, F., Nickl, R. and Spokoiny, V. (2013). Spatially adaptive density estimation by localised Haar projections., Ann. Inst. Henri Poincaré Probab. Stat. 49 900–914.
  • [13] Giné, E. and Nickl, R. (2016)., Mathematical Foundations of Infinite-dimensional Statistical Models. Cambridge University Press.
  • [14] Goldenshluger, A. and Lepski, O. (2011). Bandwidth selection in kernel density estimation: oracle inequalities and adaptive minimax optimality., Ann. Statist. 39 1608–1632.
  • [15] Härdle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998)., Wavelets, approximation, and statistical applications. Lecture Notes in Statistics 129. Springer-Verlag, New York.
  • [16] Houdré, C. and Reynaud-Bouret, P. (2003). Exponential inequalities, with constants, for U-statistics of order two. In, Stochastic inequalities and applications. Progr. Probab. 56 55–69. Birkhäuser, Basel.
  • [17] Ioannides, D. A. and Alevizos, P. D. (1997). Nonparametric regression with errors in variables and applications., Statist. Probab. Lett. 32 35–43.
  • [18] Kerkyacharian, G., Lepski, O. and Picard, D. (2001). Nonlinear estimation in anisotropic multi-index denoising., Probab. Theory Related Fields 121 137–170.
  • [19] Koo, J.-Y. and Lee, K.-W. (1998). $B$-spline estimation of regression functions with errors in variable., Statist. Probab. Lett. 40 57–66.
  • [20] Massart, P. (2007)., Concentration Inequalities and Model Selection. Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour, 6-23, 2003. Springer.
  • [21] Meister, A. (2009)., Deconvolution problems in nonparametric statistics. Lecture Notes in Statistics 193. Springer-Verlag, Berlin.
  • [22] Patel, J. K. and Read, C. B. (1982)., Handbook of the normal distribution. Statistics: Textbooks and Monographs 40. Marcel Dekker, Inc., New York.
  • [23] Whittemore, A. S. and Keller, J. B. (1988). Approximations for regression with covariate measurement error., J. Amer. Statist. Assoc. 83 1057–1066.