Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections

Bernoulli

Volumn 16, Issue 4, 2010, Pages 1137-1163

  Giné, Evarist ^a   Nickl, Richard ^b  

^a UNIVERSITY OF CONNECTICUT   (United States)

^b UNIVERSITY OF CAMBRIDGE   (United Kingdom)

Author keywords

Adaptive estimation; Lepski's method; Rademacher processes; Spline estimator; Sup norm loss; Wavelet estimator

Indexed keywords

EID: 77649293894     PISSN: 13507265     EISSN: None     Source Type: Journal    
DOI: 10.3150/09-BEJ239     Document Type: Article

References (34)

1. Barron, A., Birgé, L. and Massart, P. (1999). Risk bounds for model selection via penalization. Probab. Theory Related Fields 113 301-413.
2. Bartlett, P., Boucheron, S. and Lugosi, G. (2002). Model selection and error estimation. Mach. Learn. 48 85-113.
3. Bousquet, O. (2003). Concentration inequalities for sub-additive functions using the entropy method. In Stochastic Inequalities and Applications (E. Giné, C. Houdré and D. Nualart, eds.). Progr. Probab. 56 213-247. Boston: Birkhäuser.
4. Daubechies, I. (1992). Ten Lectures on Wavelets. CBMS-NSF Reg. Conf. Ser. in Appl. Math. 61. Philadelphia, PA: SIAM.
5. Deheuvels, P. (2000). Uniform limit laws for kernel density estimators on possibly unbounded intervals. In Recent Advances in Reliability Theory (N. Limnios and M. Nikulin, eds.) 477-492. Boston: Birkhäuser.
6. DeVore, R.A. and Lorentz, G.G. (1993). Constructive Approximation. Berlin: Springer.
7. Donoho, D.L., Johnstone, I.M., Kerkyacharian, G. and Picard, D. (1996). Density estimation by wavelet thresholding. Ann. Statist. 24 508-539.
8. Dudley, R.M. (1999). Uniform Central Limit Theorems. Cambridge: Cambridge Univ. Press.
9. Einmahl, U. and Mason, D.M. (2000). An empirical process approach to the uniform consistency of kernel-type function estimators. J. Theoret. Probab. 13 1-37.
10. Giné, E. and Guillou, A. (2001). On consistency of kernel density estimators for randomly censored data: Rates holding uniformly over adaptive intervals. Ann. Inst. H. Poincaré Probab. Statist. 37 503-522.
11. Giné, E. and Guillou, A. (2002). Rates of strong uniform consistency for multivariate kernel density estimators. Ann. Inst. H. Poincaré Probab. Statist. 38 907-921.
12. Giné, E. and Koltchinskii, V. (2006). Concentration inequalities and asymptotic results for ratio type empirical processes. Ann. Probab. 34 1143-1216.
13. Giné, E. and Nickl, R. (2009). An exponential inequality for the distribution function of the kernel density estimator, with applications to adaptive estimation. Probab. Theory Related Fields 143 569-596.
14. Giné, E. and Nickl, R. (2009). Uniform limit theorems for wavelet density estimators. Ann. Probab. 37 1605-1646.
15. Goldenshluger, A. and Lepski, O. (2009). Structural adaptation via Lp-norm oracle inequalities. Probab. Theory Related Fields 143 41-71.
16. Golubev, Y., Lepski, O. and Levit, B. (2001). On adaptive estimation for the sup-norm losses. Math. Methods Statist. 10 23-37.
17. Härdle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998). Wavelets, Approximation, and Statistical Applications. Lecture Notes in Statistics 129. New York: Springer.
18. Huang, S.-Y. (1999). Density estimation by wavelet-based reproducing kernels. Statist. Sinica 9 137-151.
19. Huang, S.-Y. and Studden, W.J. (1993). Density estimation using spline projection kernels. Comm. Statist. Theory Methods 22 3263-3285.
20. Kerkyacharian, G. and Picard, D. (1992). Density estimation in Besov spaces. Statist. Probab. Lett. 13 15-24.
21. Klein, T. and Rio, E. (2005). Concentration around the mean for maxima of empirical processes. Ann. Probab. 33 1060-1077.
22. Koltchinskii, V. (2001). Rademacher penalties and structural risk minimization. IEEE Trans. Inform. Theory 47 1902-1914.
23. Koltchinskii, V. (2006). Local Rademacher complexities and oracle inequalities in risk minimization. Ann. Statist. 34 2593-2656.
24. Korostelev, A. and Nussbaum, M. (1999). The asymptotic minimax constant for sup-norm loss in nonparametric density estimation. Bernoulli 5 1099-1118.
25. Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. Providence, RI: Amer. Math. Soc.
26. Lepski, O.V. (1991). Asymptotically minimax adaptive estimation. I. Upper bounds. Optimally adaptive estimates. Theory Probab. Appl. 36 682-697.
27. Massart, P. (2000). About the constants in Talagrand's concentration inequalities for empirical processes. Ann. Probab. 28 863-884.
28. Meyer, Y. (1992). Wavelets and Operators I. Cambridge: Cambridge Univ. Press.
29. Nolan, D. and Pollard, D. (1987). U-processes: Rates of convergence. Ann. Statist. 15 780-799.
30. Shadrin, A.Y. (2001). The L8-norm of the L2-spline projector is bounded independently of the knot sequence: A proof of de Boor's conjecture. Acta Math. 187 59-137.
31. Schumaker, L.L. (1993). Spline Functions: Basic Theory. Malabar: Krieger.
32. Talagrand, M. (1994). Sharper bounds for Gaussian and empirical processes. Ann. Probab. 22 28-76.
33. Talagrand, M. (1996). New concentration inequalities in product spaces. Invent. Math. 126 505-563.
34. Tsybakov, A.B. (1998). Pointwise and sup-norm sharp adaptive estimation of the functions on the Sobolev classes. Ann. Statist. 26 2420-2469.