Regularization of wavelet approximations

Journal of the American Statistical Association

Volumn 96, Issue 455, 2001, Pages 939-955

  Antoniadis, Anestis ^a   Fan, Jianqing ^b  

^a UNIV GRENOBLE ALPES   (France)

^b CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Asymptotic minimax; Irregular designs; Nonquadratic penality functions; Oracle inequalities; Penalized least squares; ROSE; Wavelets

Indexed keywords

EID: 0442312210     PISSN: 01621459     EISSN: 1537274X     Source Type: Journal    
DOI: 10.1198/016214501753208942     Document Type: Article

References (47)

1. Abramovich, F., Bailey, T. C., and Sapatinas, T. (2000), “Wavelet Analysis and Its Statistical Applications,” The Statistician, 49, 1–29.
2. Amato, U., and Vuza, D. T. (1997). “Wavelet Approximation of a Function From Samples Affected By Noise,” Revue Roumaine de Mathématiques Pures et Appliquées, 42, 481–493.
3. Antoniadis, A. (1996), “Smoothing Noisy Data With Tapered Coiflets Series,” Scandinavian Journal of Statistics, 23, 313–330.
4. Antoniadis, A. (1997), “Wavelets in Statistics: A Review” (with discussion), Italian Journal of Statistics, 6, 97–144.
5. Antoniadis, A., Grégoire, G., and McKeague, I. (1994), “Wavelet Methods for Curve Estimation,” Journal of the American Statistical Association, 89, 1340–1353.
6. Antoniadis, A., Grégoire, G., and Vial, P. (1997), “Random Design Wavelet Curve Smoothing,” Statistics and Probability Letters, 35, pp. 225–232.
7. Barron, A., Birgé, L., and Massart, P. (1999), “Risk Bounds for Model Selection Via Penalization,” Probability Theory and Related Fields, 113, 301–413.
8. Belge, M., Kilmer, M. E., and Miller, E. L. (2000), “Wavelet Domain Image Restoration With Adaptive Edge-Preserving Regularization,” IEEE Transactions on Image Processing, 9, 597–608.
9. Bernard, C., Mallat, S., and Slotine, J. J. (1999), “Wavelet Interpolation Networks,” Preprint, Centre de Mathématiques Appliquées, Ecole Polytechnique, France.
10. Bickel, P. J. (1975), “One-Step Huber Estimates in Linear Models,” Journal of the American Statistical Association, 70, 428–433.
11. Blake, A. (1989), “Comparison of the Efficiency of Deterministic and Stochastic Algorithms for Visual Reconstruction,” IEEE Transactions on Pattern Analysis and Machine Intelligence, 11, 2–12.
12. Blake, A., and Zisserman, A. (1987), Visual Reconstruction, Cambridge, MA: MIT Press.
13. Breiman, L. (1995), “Better Subset Regression Using the Nonnegative Garotte,” Technometrics, 37, 373–384.
14. Bouman, C., and Sauer, K. (1993), “A Generalized Gaussian Image Model for Edge-Preserving MAP Estimation,” IEEE Transactions on Image Processing, 2, 3, 296–310.
15. Cai, T. T., and Brown, L. D. (1998), “Wavelet Shrinkage for Nonequispaced Samples,” Annals of Statistics, 26, 1783–1799.
16. Chen, S. C., Donoho, D. L., and Sanders, M. A. (1998), “Atomic Decomposition by Basis Pursuit,” SIAM Journal of Scientific Computing, 20, 1, 33–61.
17. Dechevsky, L. T., and Penev, S. I. (1999), “Weak Penalized Least Squares Wavelet Regression Estimation,” Technical Report S99-1, Department of Statistics, School of Mathematics, University of New South Wales, Australia.
18. Donoho, D. L., and Johnstone, I. M. (1994), “Ideal Spatial Adaptation by Wavelet Shrinkage,” Biometrika, 81, 425–455.
19. Donoho, D. L., and Johnstone, I. M. (1998), “Minimax Estimation Via Wavelet Shrinkage,” Annals of Statistics, 26, 879–921.
20. Donoho, D. L., Johnstone, I. M., Hock, J. C., and Stern, A. S. (1992), “Maximum Entropy and the Nearly Black Object” (with discussions), Journal of the Royal Statistical Society, Ser. B, 54, 41–81.
21. Donoho, D. L., Johnstone, I. M., Kerkyacharian, G., and Picard, D. (1995), “Wavelet Shrinkage: Asymptopia?” (with discussion), Journal of the Royal Statistical Society, Ser. B, 57, 301–369.
22. Donoho, D. L., Vetterli, M., DeVore, R. A., and Daubechies, I. (1998), “Data Compression and Harmonic Analysis,” Technical report, Department of Statistics, Stanford University.
23. Fan, J. (1997), “Comment on ‘Wavelets in Statistics: A Review’ by A. Antoniadis,” Italian Journal of Statistics, 6, 97–144.
24. Fan, J., and Chen, J. (1999), “One-Step Local Quasi-Likelihood Estimation,” Journal of the Royal Statistical Society, Ser. B, 61, 927–943.
25. Fan, J., and Li, R. (1999), “Variable Selection via Penalized Likelihood,” Technical report, Department of Statistics, UCLA.
26. Gao, H. Y., and Bruce, A. G. (1997), “WaveShrink with Firm Shrinkage,” Statistica Sinica, 7, 855–874.
27. Hall, P., and Patil, P. (1995). “Formulae for Mean Integrated Squared Error of Nonlinear Wavelet-Based Density Estimators,” Annals of Statistics, 23, 905–928.
28. Hall, P., and Turlach, B. A. (1997), “Interpolation Methods for Nonlinear Wavelet Regression With Irregularly Spaced Design,” Annals of Statistics, 25, 1912–1925.
29. Hansen, P. C., and O’Leary, D. P. (1993), “The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems,” SIAM Journal of Scientific Computing, 14, 1487–1503.
30. Johnstone, I. M., and Silverman, B. W. (1997), “Wavelet Threshold Estimators for Data With Correlated Noise,” Journal of the Royal Statistical Society, Ser. B, 59, 319–351.
31. Kovac, A., and Silverman, B. W. (2000), “Extending the Scope of Wavelet Regression Methods by Coefficient-Dependent Thresholding,” Journal of the American Statistical Association, 95, 172–183.
32. McCoy, E. J. (1999). “Wavelet Regression: A Penalty Function Approach,” in Proceeding of the 52nd session of the International Statistical Institute.
33. Moulin, P., and Liu, J. (1999), “Analysis of Multiresolution Denoising Schemes Using Generalized Gaussian and Complexity Priors,” IEEE Transactions on Information Theory, 45, 9, 909–919.
34. Neumann, M. H., and Spokoiny, V. G. (1995), “On the Efficiency of Wavelet Estimators Under Arbitrary Error Distributions,” Mathematical Methods of Statistics, 4, 2, 137–166.
35. Nikolova, M. (1999), “Markovian Reconstruction Using a GNC Approach,” IEEE Transactions on Image Processing, 8, 1204–1220.
36. Nikolova, M. (1999), “Local Strong Homogeneity of a Regularized Estimator,” SIAM Journal on Applied Mathematics, 61, 633–658.
37. Nikolova, M., Idier, J., and Mohammad-Djafari, A. (in press), “Inversion of Large-Support Ill-Posed Linear Operators Using a Piecewise Gaussian MRF,” submitted to IEEE Transactions on Image Processing.
38. Ogden, T. (1997), Essential Wavelets for Statistical Applications and Data Analysis, Boston: Birkhauser.
39. Rao, C. R. (1973), Linear Statistical Inference and Its Applications (2nd ed.), New York: Wiley.
40. Robinson, P. M. (1988), “The Stochastic Difference Between Econometric and Statistics,” Econometrica, 56, 531–547.
41. Sardy, S., Percival, D. B., Bruce A. G., Gao, H.-Y., and Stuelzle, W. (1999), “Wavelet Shrinkage for Unequally Spaced Data,” Statistics and Computing, 9, 65–75.
42. Silverman, B. W. (1985), “Some Aspects of the Spline Smoothing Approach to Nonparametric Regression Curve Fitting” (with discussion), Journal of the Royal Statistical Society, Ser. B, 47, 1–52.
43. Solo, V. (1998), “Wavelet Signal Estimation in Coloured Noise With Extension to Transfer Function Estimation,” in Proceedings of the 37th IEEE Conference on Decision and Control, 4, 3940–3941.
44. Tibshirani, R. (1995), “Regression Shrinkage and Selection via the Lasso,” Journal of the Royal Statistical Society, Ser. B, 57, 267–288.
45. Vidakovic, B. (1999), Statistical Modeling by Wavelets. New York: Wiley.
46. Vogel, C. R. (1996), “Non-Convergence of the L-Curve Regularization Parameter Selection Method,” Inverse Problems, 12, 535–547.
47. Wang, Y. (1996), “Function Estimation via Wavelet Shrinkage for Long-Memory Data,” Annals of Statistics, 24, 466–484.