## Adaptive estimation with soft thresholding penalties.(English)Zbl 1090.62534

Summary: We show that various robust nonparametric regression estimators, such as the least absolute deviations estimator, can be made adaptive (up to logarithmic factors), by adding a soft thresholding type penalty to the loss function. As an example, we consider the situation where the roughness of the regression function is described by a single parameter $$p$$. The theory is complemented with a simulation study.

### MSC:

 62G08 Nonparametric regression and quantile regression 62G35 Nonparametric robustness
Full Text:

### References:

 [1] Baraud Y., Probability Theory and Related Fields 117 pp 467– (2000) [2] Barron A., The Annals of Statistics 19 pp 1347– (1991) [3] DOI: 10.1007/s004400050210 · Zbl 0946.62036 [4] Bergh J., Interpolation spaces - an introduction (1976) · Zbl 0344.46071 [5] Besov O., Integral representations of functions and embedding theorems, Vol. I (1978) [6] L. Birge, P. Massart, D. Pullard, E. Torgerson, and G. Yang (1997 ), From model selection to adaptive estimation , in: (eds.), Festschrift for Lucien Le Cam: Research Papers in Probability and Statistics, Springer, New York, 55 -87 . [7] DOI: 10.1007/s003659910001 · Zbl 1004.41006 [8] DOI: 10.1007/s100970100031 · Zbl 1037.62001 [9] Birman M., Matematisheskii Sbornik (Noveaya Seriya) 73 pp 331– (1967) [10] A. Bruce, S. Sardy, and P. Tseng (1999 ), Robust wavelet denoising , Preprint. [11] Chen S., SIAM Journal of Scientific Computing 20 pp 33– (1999) [12] DeVore R., Constructive approximation (1993) [13] DOI: 10.1109/18.382009 · Zbl 0820.62002 [14] Donoho D., Probability Theory and Related Fields 99 pp 277– (1994) [15] Donoho D., Biometrika 81 pp 425– (1994) [16] Donoho D., Bernoulli 2 pp 39– (1996) [17] D. Donoho, I. Johnstone, G. Kerkacharian, D. Picard, D. Pullard, E. Torgerson, and G. Yang (1996 ), Universal near minimaxity of wavelet shrinkage , in: (eds.), Festschrift for Lucien Le Cam: Research Papers in Probability and Statistics, Springer, New York, 183 -218 . [18] DOI: 10.1214/aos/1032894451 · Zbl 0860.62032 [19] Edmunds E., Proceedings of the London Mathematical Society 64 pp 153– (1992) [20] Hoeffding W., Journal of the American Statistical Association 58 pp 13– (1963) [21] Ledoux M., Probability in Banach spaces, isoperimetry and processes (1991) · Zbl 0748.60004 [22] Mallat S., A wavelet tour of signal processing (1998) · Zbl 0937.94001 [23] DOI: 10.1214/aos/1034276635 · Zbl 0871.62040 [24] Massart P., Annales de la Faculte des Sciences de Toulouse 9 pp 245– (2000) · Zbl 0986.62002 [25] Meyer Y., Contributions to nonlinear partial differential equations, Vol. II (Paris, 1985) pp 158– (1987) [26] DOI: 10.1214/aos/1034276636 · Zbl 0898.62044 [27] Rockafellar R., Convex analysis (1997) [28] Silverman B., The Annals of Statistics 10 pp 795– (1982) [29] Stone C., The Annals of Statistics 18 pp 717– (1990) [30] Tibshirani R., Journal of the Royal Statistical Society B 58 pp 267– (1996) [31] Geer S., The Annals of Statistics 18 pp 907– (1990) [32] Geer S., Journal of Statistical Planning and Inference pp 31– (1999) [33] Geer S., Empirical processes in M-estimation (2000) [34] Vaart A., Weak convergence and empirical processes, with applications to statistics (1996) · Zbl 0862.60002 [35] Yang Y., Statistica Sinica 9 pp 475– (1999)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.