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Statistics and Computing

, Volume 17, Issue 4, pp 293–310 | Cite as

Robust estimation and wavelet thresholding in partially linear models

  • Irène GannazEmail author
Article

Abstract

This paper is concerned with a semiparametric partially linear regression model with unknown regression coefficients, an unknown nonparametric function for the non-linear component, and unobservable Gaussian distributed random errors. We present a wavelet thresholding based estimation procedure to estimate the components of the partial linear model by establishing a connection between an l 1-penalty based wavelet estimator of the nonparametric component and Huber’s M-estimation of a standard linear model with outliers. Some general results on the large sample properties of the estimates of both the parametric and the nonparametric part of the model are established. Simulations are used to illustrate the general results and to compare the proposed methodology with other methods available in the recent literature.

Keywords

Semi-nonparametric models Partially linear models Wavelet thresholding Backfitting M-estimation Penalized least-squares 

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References

  1. Antoniadis, A., Fan, J.: Regularization of wavelet approximations. J. Am. Stat. Assoc. 96, 939–967 (2001) zbMATHCrossRefGoogle Scholar
  2. Bai, Z., Rao, C., Wu, Y.: M-estimation of multivariate linear regression parameters under a convex discrepancy function. Stat. Sinica 2, 237–254 (1992) zbMATHGoogle Scholar
  3. Chang, X., Qu, L.: Wavelet estimation of partially linear models. Comput. Stat. Data Anal. 47, 31–48 (2004) CrossRefGoogle Scholar
  4. Charbonnier, P., Blanc-Feraud, G., Barlaud, M.: Deterministic edge-preserving regularization in computed imaging. Trans. Image Process. 6, 298–311 (1997) CrossRefGoogle Scholar
  5. Chen, H.: Estimation of semiparametric generalized linear models. Technical Report, State University of New York (1987) Google Scholar
  6. Chen, H.: Convergence rates for parametric components in a partly linear model. Ann. Stat. 16, 136–146 (1988) zbMATHGoogle Scholar
  7. Chen, H., Chen, K.-W.: Selection of the splined variables and convergence rates in a partial spline model. Can. J. Stat. 19, 323–339 (1991) zbMATHCrossRefGoogle Scholar
  8. Chen, H., Shiau, J.-J.H.: A two-stage spline smoothing method for partially linear models. J. Stat. Plan. Inference 27, 187–201 (1991) zbMATHCrossRefGoogle Scholar
  9. Dahyot, R., Kokaram, A.: Comparison of two algorithms for robust M-estimation of global motion parameters. http://citeseer.ist.psu.edu/709403.html (2004)
  10. Dahyot, R., Charbonnier, P., Heitz, F.: A Bayesian approach to object detection using probabilistic appearance-based models. Pattern Anal. Appl. 7, 317–332 (2004) Google Scholar
  11. Donald, S., Newey, W.: Series estimation of semilinear models. J. Multivar. Anal. 50, 30–40 (1994) zbMATHCrossRefGoogle Scholar
  12. Donoho, D.: De-noising by soft-thresholding. Technical Report, Department of Statistics, Stanford University (1992) Google Scholar
  13. Donoho, D.L., Johnstone, I.M.: Minimax estimation via wavelet shrinkage. Ann. Stat. 26, 879–921 (1998) zbMATHCrossRefGoogle Scholar
  14. Donoho, D., Johnstone, I., Kerkyacharian, G., Picard, D.: Wavelet shrinkage: asymptotia? J. Roy. Stat. Soc. 57, 301–369 (1995) zbMATHGoogle Scholar
  15. Engle, R., Granger, C., Rice, J., Weiss, A.: Semiparametric estimates of the relation between weather and electricity sales. J. Am. Stat. Assoc. 81, 310–320 (1986) CrossRefGoogle Scholar
  16. Fadili, J., Bullmore, E.: Penalized partially linear models using sparse representation with an application to fMRI time series. IEEE Trans. Signal Process. 53, 3436–3448 (2005) CrossRefGoogle Scholar
  17. Geman, D., Reynolds, G.: Constrained restoration and the recovery of discontinuities. IEEE Trans. Pattern Anal. Mach. Intell. 14, 367–383 (1992) CrossRefGoogle Scholar
  18. Geman, D., Yang, C.: Nonlinear image recovery with half-quadratic regularization. IEEE Trans. Image Process. 4, 932–946 (1995) CrossRefGoogle Scholar
  19. Green, P., Yandell, B.: Semi-parametric generalized linear models. Technical Report No. 2847, University of Wisconsin-Madison (1985) Google Scholar
  20. Hamilton, S., Truong, Y.: Local estimation in partly linear models. J. Multivar. Anal. 60, 1–19 (1997) zbMATHCrossRefGoogle Scholar
  21. Hampel, F.R., Rousseeuw, P.J., Ronchetti, E., Stahel, W.A.: Robust Statistics: The Approach Based on Influence Functions. Wiley Series in Probability and Mathematical Statistics (1986) Google Scholar
  22. Hardle, W., Liang, H., Gao, J.: Partially Linear Models. Springer, New York (2000) Google Scholar
  23. Huber, P.: Robust Statistics. Wiley Series in Probability and Mathematical Statistics (1981) Google Scholar
  24. Mallat, S.: A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989) zbMATHCrossRefGoogle Scholar
  25. Mallat, S.: A Wavelet Tour on Signal Processing, 2 edn. Academic, New York (1999) Google Scholar
  26. Meyer, F.: Wavelet-based estimation of a semiparametric generalized linear model of fMRI time-series. IEEE Trans. Med. Imaging 22, 315–324 (2003) CrossRefGoogle Scholar
  27. Nikolova, M., Ng, M.: Analysis of half-quadratic minimization methods for signal and image recovery. SIAM J. Sci. Comput. 27, 937–966 (2005) zbMATHCrossRefGoogle Scholar
  28. Rice, J.: Convergence rates for partially splined models. Stat. Probab. Lett. 4, 203–208 (1986) zbMATHCrossRefGoogle Scholar
  29. Rockafellar, R.: Convex Analysis. Princeton University Press, Princeton (1970) zbMATHGoogle Scholar
  30. Schick, A.: Root-n-consistent and efficient estimation in semiparametric additive regression models. Stat. Probab. Lett. 30, 45–51 (1996) zbMATHCrossRefGoogle Scholar
  31. Speckman, P.: Kernel smoothing in partial linear models. J. Roy. Stat. Soc. 50, 413–436 (1988) zbMATHGoogle Scholar
  32. Vik, T.: Modèles statistiques d’apparence non gaussiens. Application à la création d’un atlas probabiliste de perfusion cérebrale en imagerie médicale. Ph.D. disertation, Université Strasbourg 1 (2004) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Laboratoire de Modélisation et CalculUniversité Joseph FourierGrenoble Cedex 9France

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