Geodesic
Upper Bounds for Errors of Estimators in a Problem of Nonparametric Regression: The Adaptive Case and the Case of Unknown Measure~$\rho_X$
Matematičeskie zametki, Tome 86 (2009) no. 5, pp. 725-732.

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We construct estimators of regression functions and prove theorems on their errors in two different cases. In the first case, we consider the so-called adaptive estimators whose error is close to the optimal one for a whole family of classes of possible regression functions; the adaptivity of the estimators is connected with the fact that they are constructed without any information about the choice of the class. In the second case, the class of possible regression functions is fixed; however, the marginal measure is unknown and the estimator is constructed without any information about this measure. Its error turns out to be close to the minimal possible (in the worst case) error.
Keywords: nonparametric regression, regression function, adaptive estimator, marginal measure, Bernstein's inequality, combinatorial dimension, least-squares method. 
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Yu. V. Malykhin. Upper Bounds for Errors of Estimators in a Problem of Nonparametric Regression: The Adaptive Case and the Case of Unknown Measure~$\rho_X$. Matematičeskie zametki, Tome 86 (2009) no. 5, pp. 725-732. http://geodesic.mathdoc.fr/item/MZM_2009_86_5_a8/

[1] F. Cucker, S. Smale, “On the mathematical foundations of learning”, Bull. Amer. Math. Soc. (N.S.), 39:1 (2002), 1–49 | DOI | MR | Zbl

[2] R. DeVore, G. Kerkyacharian, D. Picard, V. Temlyakov, “Approximation methods for supervised learning”, Found. Comput. Math., 6:1 (2006), 3–58 | DOI | MR | Zbl

[3] V. N. Temlyakov, “Approximation in learning theory”, Constr. Approx., 27:1 (2008), 33–74 | DOI | MR | Zbl

[4] S. V. Konyagin, E. D. Livshits, “Ob adaptivnykh otsenschikakh v statisticheskoi teorii obucheniya”, Teoriya funktsii i nelineinye uravneniya v chastnykh proizvodnykh, Sbornik statei. K 70-letiyu so dnya rozhdeniya chlena-korrespondenta RAN Stanislava Ivanovicha Pokhozhaeva, Tr. MIAN, 260, MAIK, M., 2008, 193–201 | MR

[5] L. Györfi, M. Kohler, A. Krzyżak, W. Harro, A Distribution-Free Theory of Nonparametric Regression, Springer Ser. Statist., Springer-Verlag, New York, NY, 2002 | DOI | MR | Zbl

[6] S. Mendelson, R. Vershynin, “Entropy and the combinatorial dimension”, Invent. Math., 152:1 (2003), 37–55 | DOI | MR | Zbl

[7] M. Rudelson, R. Vershynin, “Combinatorics of random processes and sections of convex bodies”, Ann. of Math. (2), 164:2 (2006), 603–648 | DOI | MR | Zbl