Optimal rates and adaptation in the single-index model using aggregation

Electronic Journal of Statistics

Volumn 1, Issue , 2007, Pages 538-573

  Gaïffas, Stéephane ^a   Lecué, Guillaume ^a  

^a SORBONNE UNIVERSITÉ   (France)

Author keywords

Adaptation; Aggregation; Minimax; Nonparametric regression; Oracle inequality; Random design; Single index

Indexed keywords

EID: 70350021658     PISSN: 19357524     EISSN: None     Source Type: Journal    
DOI: 10.1214/07-EJS077     Document Type: Article

References (27)

1. Audibert, J.-Y. and Tsybakov, A. (2007). Fast learning rates for plug-in esti-mators under the margin condition. The Annals of Statistics, 35.
2. Birgé, L. (2005). Model selection via testing: an alternative to (penalized) maximum likelihood estimators. Available at http://www.proba.jussieu.fr/mathdoc/textes/PMA-862.pdf.
3. Catoni, O. (2001). Statistical Learning Theory and Stochastic Optimization. Ecole d’été de Probabilités de Saint-Flour 2001, Lecture Notes in Mathematcs, Springer, N.Y.
4. Delecroix, M., Härdle, W. and Hristache, M. (2003). Efficient estimation in conditional single-index regression. J. Multivariate Anal., 86 213-226.
5. Delecroix, M., Hristache, M. and Patilea, V. (2006). On semiparametric M-estimation in single-index regression. J. Statist. Plann. Inference, 136 730-769.
6. Fan, J. and Gijbels, I. (1995). Data-driven bandwidth selection in local polynomial fitting: variable bandwidth and spatial adaptation. Journal of the Royal Statistical Society. Series B. Methodological, 57 371-394.
7. Fan, J. and Gijbels, I. (1996). Local polynomial modelling and its applications. Monographs on Statistics and Applied Probability, Chapman & Hall, London.
8. Geenens, G. and Delecroix, M. (2005). A survey about single-index models theory. URL http://www.stat.ucl.ac.be/ISpub/dp/2005/dp0508.pdf.
9. Györfi, L., Kohler, M., Krzyżak, A. and Walk, H. (2002). A distribution-free theory of nonparametric regression. Springer Series in Statistics, SpringerVerlag, New York.
10. Horowitz, J. L. (1998). Semiparametric methods in econometrics, vol. 131 of Lecture Notes in Statistics. Springer-Verlag, New York.
11. Hristache, M., Juditsky, A. and Spokoiny, V. (2001). Direct estimation of the index coefficient in a single-index model. Ann. Statist., 29 595-623.
12. Juditsky, A. and Nemirovski, A. (2000). Functional aggregation for non-parametric estimation. Ann. Statist., 28(3) 681-712.
13. Juditsky, A., Rigollet, P. and Tsybakov, A. (2005a). Learning by mirror averaging. URL http://arxiv.org/abs/math/0511468.
14. Juditsky, A. B., Nazin, A. V., Tsybakov, A. B. and Vayatis, N. (2005b). Recursive aggregation of estimators by the mirror descent method with averaging. Problemy Peredachi Informatsii, 41 78-96.
15. Lecué, G. (2007). Suboptimality of penalized empirical risk minimization in classification. 20th Annual Conference On Learning Theory, COLT07, Springer, 142-156.
16. Leung, G. and Barron, A. R. (2006). Information theory and mixing least-squares regressions. IEEE Trans. Inform. Theory, 52 3396-3410.
17. Nemirovski, A. (2000). Topics in Non-parametric Statistics, vol. 1738 of Ecole d’été de Probabilités de Saint-Flour 1998, Lecture Notes in Mathematics. Springer, N.Y.
18. Stone, C. J. (1982). Optimal global rates of convergence for nonparametric regression. The Annals of Statistics, 10 1040-1053.
19. Stute, W. and Zhu, L.-X. (2005). Nonparametric checks for single-index models. Ann. Statist., 33 1048-1083.
20. Tsybakov, A. (2003a). Introduction l’estimation non-paramerique. Springer.
21. Tsybakov, A. B. (2003b). Optimal rates of aggregation. Computational Learning Theory and Kernel Machines. B.Schölkopf and M.Warmuth, eds. Lecture Notes in Artificial Intelligence, 2777 303-313. Springer, Heidelberg.
22. Vapnik, V. N. (1998). Statistical learning theory. Adaptive and Learning Systems for Signal Processing, Communications, and Control, John Wiley & Sons Inc., New York., A Wiley-Interscience Publication.
23. Xia, Y. and Härdle, W. (2006). Semi-parametric estimation of partially linear single-index models 1162-1184.
24. Yang, Y. (2000a). Combining different procedures for adaptive regression. J. Multivariate Anal., 74 135-161.
25. Yang, Y. (2000b). Mixing strategies for density estimation. Ann. Statist., 28 75-87.
26. Yang, Y. (2004). Aggregating regression procedures to improve performance. Bernoulli, 10 25-47.
27. Yang, Y. and Barron, A. (1999). Information-theoretic determination of minimax rates of convergence. Ann. Statist., 27 1564-1599.