Structural adaptation via double-struck Lp-norm oracle inequalities

Probability Theory and Related Fields

Volumn 143, Issue 1-2, 2009, Pages 41-71

  Goldenshluger, Alexander ^a   Lepski, Oleg ^b  

^a UNIVERSITY OF HAIFA   (Israel)

^b AIX MARSEILLE UNIVERSITÉ   (France)

Author keywords

Adaptive estimation; Minimax risk; Optimal rates of convergence; Oracle inequalities; Structural adaptation

Indexed keywords

EID: 54349118425     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00440-007-0119-5     Document Type: Article

References (34)

1. A. Barron L. Birgé P. Massart 1999 Risk bounds for model selection via penalization Probab. Theory Relat. Fields 113 301 413
2. Belomestny, D., Spokoiny, V.: Local likelihood modeling via stagewise aggregation. WIAS preprint No. 1000 (2004). www.wias-berlin.de
3. K. Bertin 2004 Asymptotically exact minimax estimation in sup-norm for anisotropic Hölder balls Bernoulli 10 873 888
4. L. Cavalier G.K. Golubev D. Picard A.B. Tsybakov 2002 Oracle inequalities for inverse problems Ann. Stat. 30 843 874
5. H. Chen 1991 Estimation of a projection-pursuit type regression model Ann. Stat. 19 142 157
6. Devroye, L., Lugosi, G.: Combinatorial Methods in Density Estimation. Springer, New York (2001)
7. Folland, G.B.: Real Analysis, 2nd edn. Wiley, New York (1999)
8. A. Goldenshluger A. Nemirovski 1997 On spatially adaptive estimation of nonparametric regression Math. Methods Stat. 6 135 170
9. G.K. Golubev 1992 Asymptotically minimax estimation of a regression function in an additive model Probl. Inform. Transm. 28 101 112
10. G.K. Golubev 2004 The method of risk envelopes in the estimation of linear functionals (Russian) Probl. Inform. Transm. 40 53 65
11. Györfi, L., Kohler, M., Krzyzak, A., Walk, H.: A Distribution-Free Theory of Nonparametric Regression. Springer, New York (2002)
12. P. Hall 1989 On projection-pursuit regression Ann. Stat. 17 573 588
13. M. Hristache A. Juditsky V. Spokoiny 2001 Direct estimation of the index coefficient in a single-index model Ann. Stat. 29 595 623
14. M. Hristache A. Juditsky J. Polzehl V. Spokoiny 2001 Structure adaptive approach for dimension reduction Ann. Stat. 29 1537 1566
15. P. Huber 1985 Projection pursuit. With discussion Ann. Stat. 13 435 525
16. I.A. Ibragimov R.Z. Khasminskii 1982 Bounds for the quality of nonparametric estimation of regression Theory Probab. Appl. 27 81 94
17. I.A. Ibragimov 2004 Estimation of multivariate regression Theory Probab. Appl. 48 256 272
18. R. Jennrich 1969 Asymptotic properties of non-linear least squares estimators Ann. Math. Stat. 40 633 643
19. Johnstone, I.M.: Oracle inequalities and nonparametric function estimation. In: Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998). Doc. Math., Extra vol. III, pp. 267-278 (1998)
20. Juditsky, A., Lepski, O., Tsybakov, A.: Statistical estimation of composite functions. Manuscript (2006)
21. G. Kerkyacharian O. Lepski D. Picard 2001 Nonlinear estimation in anisotropic multi-index denoising Probab. Theory Relat. Fields 121 137 170
22. O.V. Lepski B.Y. Levit 1999 Adaptive nonparametric estimation of smooth multivariate functions Math. Methods Stat. 8 344 370
23. O. Lepski E. Mammen V. Spokoiny 1997 Optimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimators with variable bandwidth selectors Ann. Stat. 25 929 947
24. O.V. Lepski V.G. Spokoiny 1997 Optimal pointwise adaptive methods in nonparametric estimation Ann. Stat. 25 2512 2546
25. Lifshits, M.: Gaussian Random Functions. Kluwer, Dordrecht (1995)
26. Nemirovski, A.S.: Nonparametric estimation of smooth regression functions. Soviet J. Comput. Syst. Sci. 23(6), 1-11 (1985);
27. translated from Izv. Akad. Nauk SSSR Tekhn. Kibernet. 235 (3), 50-60 (1985)(Russian)
28. Nemirovski, A.: Topics in Non-parametric Statistics. Lectures on probability theory and statistics (Saint-Flour, 1998), Lecture Notes in Mathematics, vol. 1738, pp. 85-277. Springer, Berlin (2000)
29. T. Nicoleris Y. Yatracos 1997 Rates of convergence of estimators, Kolmogorov's entropy and the dimensionality reduction principle in regression Ann. Stat. 25 2493 2511
30. k Theory Probab. Appl. 31 108 115
31. C.J. Stone 1982 Optimal global rates of convergence for nonparametric regression Ann. Stat. 10 1040 1053
32. C.J. Stone 1985 Additive regression and other nonparametric models Ann. Stat. 13 689 705
33. M. Talagrand 1994 Sharper bounds for Gaussian and empirical processes Ann. Probab. 22 28 76
34. Tsybakov, A.: Optimal rates of aggregation. In: Scholkopf, B., Warmuth, M. (eds) Computational Learning Theory and Kernel Machines. Lectures Notes in Artificial Intelligence, vol.2777, pp. 303-313. Springer, Heidelberg (2003)