Adaptation to anisotropy and inhomogeneity via dyadic piecewise polynomial selection

Mathematical Methods of Statistics

Volumn 21, Issue 1, 2012, Pages 1-28

  Akakpo, N ^a  

^a SORBONNE UNIVERSITÉ   (France)

Author keywords

anisotropic Besov spaces; density estimation; multivariate estimation; nonlinear approximation; spatial adaptation

Indexed keywords

EID: 84870375311     PISSN: 10665307     EISSN: 19348045     Source Type: Journal    
DOI: 10.3103/S1066530712010012     Document Type: Article

References (29)

1. N. Akakpo and C. Durot, "Histogram Selection for Possibly Censored Data", Math. Methods Statist. 19(3), 189-218 (2010).
2. S. Boucheron, O. Bousquet, G. Lugosi, and P. Massart, "Moment Inequalities for Functions of Independent Random Variables", Ann. Probab. 33(2), 514-560 (2005).
3. A. Barron, L. Birgé, and P. Massart, "Risk Bounds for Model Selection via Penalization", Probab. Theory Rel. Fields 113(3), 301-413 (1999).
4. L. Birgé, Model Selection via Testing: An Alternative to (Penalized) Maximum Likelihood Estimators, Ann. Inst. H. Poincaré Probab. Statist. 42(3), 273-325 (2006).
5. L. Birgé, "Statistical Estimation withModel Selection", Indag. Math. (N. S.) 17(4), 497-537 (2006).
6. L. Birgé and P. Massart, "From Model Selection to Adaptive Estimation", in Festschrift for Lucien Le Cam (Springer, New York, 1997), pp. 55-87.
7. L. Birgé and P. Massart, "An Adaptive Compression Algorithm in Besov Spaces", Constr. Approx. 16(1), 1-36 (2000).
8. G. Blanchard, C. Schäfer, and Y. Rozenholc, "Oracle Bounds and Exact Algorithm for Dyadic Classificatdion Trees", in Learning Theory (2004), pp. 378-392.
9. G. Blanchard, C. Schäfer, Y. Rozenholc, and K. R. Müller. "Optimal Dyadic Decision Trees", Machine Learning 66(2), 209-241 (2007).
10. A. Cohen, W. Dahmen, I. Daubechies, and R. DeVore, "Tree Approximation and Optimal Encoding", Appl. Comput. Harmon. Anal. 11(2), 192-226 (2001).
11. A. Cohen and J. M. Mirebeau, "Adaptive and Anisotropic Piecewise Polynomial Approximation", in Multiscale, Nonlinear and Adaptive Approximation (2009), pp. 75-135.
12. F. Comte and Y. Rozenholc. "A New Algorithm for Fixed Design Regression and Denoising", Ann. Inst. Statist. Math. 56(3), 449-473 (2004).
13. R. A. DeVore, B. Jawerth, and V. Popov, "Compression ofWavelet Decompositions", Amer. J. Math. 114(4), 737-785 (1992).
14. D. L. Donoho, "CART and Best-Ortho-Basis: A Connection", Ann. Statist. 25(5), 1870-1911 (1997).
15. R. A. DeVore and X. M. Yu. "Degree of Adaptive Approximation", Math. Comp. 55(192), 625-635 (1990).
16. A. Goldenshluger and O. Lepski, "Bandwidth Selection in Kernel Density Estimation: Oracle Inequalities and Adaptive Minimax Optimality", Ann. Statist. 39(3), 1608-1632 (2011).
17. R. Hochmuth, "Wavelet Characterizations for Anisotropic Besov Spaces", Appl. Comput. Harmon. Anal. 12(2), 179-208 (2002).
18. R. Hochmuth, "n-Term Approximation in Anisotropic Function Spaces", Math. Nachr. 244, 131-149 (2002).
19. J. Klemelä, "Multivariate Histograms with Data-Dependent Partitions", Statist. Sinica 19(1), 159-176 (2009).
20. G. Kerkyacharian, O. Lepski, and D. Picard, "Nonlinear Estimation in AnisotropicMulti-Index Denoising", Probab. Theory Rel. Fields 121(2), 137-170 (2001).
21. C. Leisner, "NonlinearWavelet Approximation in Anisotropic Besov Spaces", Indiana Univ. Math. J. 52(2), 437-455 (2003).
22. P. Massart, Concentration Inequalities and Model Selection, in Lecture Notes in Mathematics (Springer, Berlin, 2007), Vol. 1896; Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour, July 6-23, 2003, with a foreword by Jean Picard.
23. M. H. Neumann, "Multivariate Wavelet Thresholding in Anisotropic Function Spaces", Statist. Sinica 10(2), 399-431 (2000).
24. M. H. Neumann and R. von Sachs, "Wavelet Thresholding in Anisotropic Function Classes and Application to Adaptive Estimation of Evolutionary Spectra", Ann. Statist. 25(1), 38-76 (1997).
25. P. Reynaud-Bouret, V. Rivoirard, and C. Tuleau-Malot, "Adaptive Density Estimation: A Curse of Support?" J. Statist. Plann. Inference, 2010.
26. H.-J. Schmeisser and H. Triebel, Topics in Fourier Analysis and Function Spaces, in Mathematik und ihre Anwendungen in Physik und Technik [Mathematics and its Applications in Physics and Technology] (Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1987), Vol. 42.
27. H. Triebel, "Entropy Numbers in Function Spaces with Mixed Integrability", Revista Matemática Complutense 24(1), 169-188 (2011).
28. R. M. Willett and R. D. Nowak, "Multiscale Poisson Intensity and Density Estimation", IEEE Trans. Inform. Theory 53(9), 3171-3187 (2007).
29. Y. Yang and A. Barron, "Information-Theoretic Determination of Minimax Rates of Convergence", Ann. Statist. 27(5), 1564-1599 (1999).