A new algorithm for fixed design regression and denoising

Annals of the Institute of Statistical Mathematics

Volumn 56, Issue 3, 2004, Pages 449-473

  Comte, F ^a   Rozenholc, Y ^b,c  

^a UNIVERSITÉ PARIS DESCARTES   (France)

^b UNIVERSITÉ PARIS DESCARTES   (France)

^c UNIVERSITÉ DU MAINE   (France)

Author keywords

Adaptive estimation; Algorithm for denoising; Dynamical programmation; Least squares regression; Model selection; Piecewise polynomials

Indexed keywords

FUNCTIONS; POLYNOMIALS; PROBLEM SOLVING; RANDOM PROCESSES; REGRESSION ANALYSIS; WAVELET TRANSFORMS;

ADAPTIVE ESTIMATIONS; ALGORITHM FOR DENOISING; DYNAMICAL PROGRAMMATION; LEAST-SQUARES REGRESSION; PIECEWISE POLYNOMIALS;

ALGORITHMS;

EID: 11144275564     PISSN: 00203157     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02530536     Document Type: Article

References (37)

1. Abramowitz, A. and Stegun, I. A. (1972). Handbook of Mathematical Functions, Dover Publications, New York.
2. Antoniadis, A. and Fan, J. (2001). Regularization of wavelets approximations (with discussion), Journal of the American Statistical Association, 96, 939-967.
3. Antoniadis, A. and Pham, D. T. (1998). Wavelet regression for random or irregular design, Computational Statistics and Data Analysis, 28, 353-369.
4. Antoniadis, A., Bigot, J. and Sapatinas, T. (2002). Wavelet estimators in nonparametric regression: A comparative simulation study, Journal of Statistical Software, 6 (see http://www.jstatsoft.org/v06/i06).
5. Baraud, Y. (2000). Model selection for regression on a fixed design, Probability Theory and Related Fields, 117, 467-493.
6. Baraud, Y. (2002). Model selection for regression on a random design, ESAIM Probability and Statistics, 6, 127-146.
7. Baraud, Y., Comte, F. and Viennet, G. (2001a). Adaptive estimation in an autoregressive and a geometrical β-mixing regression framework, The Annals of Statistics, 39, 839-875.
8. Baraud, Y., Comte, F. and Viennet, G. (2001b). Model Selection for (auto-)regression with dependent data, ESAIM Probability and Statistics, 5, 33-49.
9. Barron, A. and Cover, T. M. (1991). Minimum complexity density estimation, IEEE Transactions on Information Theory, 37, 1037-1054.
10. Barron, A., Birgé, L. and Massart, P. (1999). Risks bounds for model selection via penalization, Probability Theory and Related Fields, 113, 301-413.
11. Birgé, L. and Massart, P. (1998). Minimum contrast estimators on sieves: Exponential bounds and rates of convergence, Bernoulli, 4, 329-375.
12. Birgé, L. and Massart, P. (2001). Gaussian model selection, Journal of the European Mathematical Society, 3, 203-268.
13. Birgé, L. and Rozenholc, Y. (2002). How many bins should be put in a regular histogram. Preprint du LPMA 721, http://www.proba.jussieu.fr/mathdoc/preprints/index.html.
14. Breimann, L., Friedman, J. H., Olshen, R. H. and Stone, C. J. (1984). Classification and Regression Trees, Wadsworth, Belmont, California.
15. Buckheit, J. B., Chen, S., Donoho, D. L., Johnstone, I. M. and Scargle, J. (1995). About WaveLab, Tech. Report, Department of Statistics, Stanford University, Stanford, California, available http://www-stat.stanford.edu/wavelab
16. Cai, T. T. (1999). Adaptive wavelet estimation: A block thresholding and oracle inequality approach, The Annals of Statistics, 27, 898-924.
17. Cai, T. T. and Silverman, B. W. (2001). Incorporating information on neighboring coefficients into wavelet estimation, Sankhya, Series B, 63, 127-148.
18. Castellan, G. (2000). Sélection d'histogrammes à l'aide d'un critère de type Akaike (Histograms selection with an Akaike type criterion), Comptes Rendus de l'Académie des Sciences. Paris. Série I. Mathematique, 330, 729-732.
19. Coifman, R. R. and Donoho, D. L. (1995). Translation-invariant de-noising (eds. Antoniadis, A. and Oppenheim, G.), Wavelets and Statistics, Lecture Notes in Statistics, 103, 125-150, Springer-Verlag, New York.
20. Comte, F. and Rozenholc, Y. (2002). Adaptive estimation of mean and volatility functions in (auto)-regressive models, Stochastic Processes and Their Applications, 97, 111-145.
21. Denison, D. G. T., Mallick, B. K. and Smith, A. F. M. (1998). Automatic Bayesian curve fitting, Journal of the Royal Statistical Society Series B, 60, 333-350.
22. Donoho, D. L. (1995). Denoising by soft-thresholding, IEEE Transactions on Information Theory, 41, 613-627.
23. Donoho, D. L. and Johnstone, I. M. (1994). Ideal space adaptation by wavelet shrinkage, Biometrika, 81, 425-455.
24. Donoho, D. L., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1995). Wavelet shrinkage: Asymptopia (with discussion), Journal of the Royal Statistical Society Series B, 57, 371-394.
25. Efromovich, S. and Pinsker, M. (1984). Learning algorithm for nonparametric filtering, Automatic Remote Control, 11, 1434-1440.
26. Friedman, J. H. and Silverman, B. W. (1989). Flexible parsimonious smoothing and additive modeling, Technometrics, 31, 3-39.
27. Hastie, T. J. and Tibshirani, R. J. (1990). Generalized Additive Models, Chapman-Hall, London.
28. Huang, S. Y. and Lu, H. H.-S. (2000). Bayesian wavelet shrinkage for nonparametric mixed-effects models, Statistica Sinica, 10, 1021-1040.
29. Kanazawa, Y. (1992). An optimal variable cell histogram based on the sample spacings, The Annals of Statistics, 20, 291-304.
30. Kohler, M. (1999). Nonparametric estimation of piecewise smooth regression functions, Statistics and Probability Letters, 43, 49-55.
31. l cross-validation and generalized cross-validation: Discrete index set, The Annals of Statistics, 15, 958-975.
32. Lindstrom, M. J. (1999). Penalized estimation of free-knot splines, Journal of Computational and Graphical Statistics, 8, 333-352.
33. p, Technometrics, 15, 661-675.
34. Misiti, M., Oppenheim, G. and Poggi, J.-M. (1995). The Wavelet Toolbox (ed. The MathWorks).
35. p test for the orthogonal series estimation of regression, Theory of Probability and Its Applications, 35, 293-306.
36. Shibata, R. (1981). An optimal selection of regression variables, Biometrika, 68, 45-54.
37. Vidakovic, B. and Ruggeri, F. (2001). BAMS method: Theory and simulations, Special Issue on Wavelets, Sankhya Series B, 63(2), 234-249.