THÈSE DE DOCTORAT DE L'UNIVERSITÉ PARIS 6 Spécialité ...

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Bibliographie 125Micchelli, C. A. and Rivlin, T. J. (1977). A survey of optimal recovery. In Optimal estimationin approximation theory (Proc. Internat. Sympos., Freudenstadt, 1976), pages1–54. Plenum, New York.Nemirovski, A. (2000). Topics in non-parametric statistics. In Lectures on probabilitytheory and statistics (Saint-Flour, 1998), volume 1738 of Lecture Notes in Math., pages85–277. Springer, Berlin.Neumann, M. H. and von Sachs, R. (1997). Wavelet thresholding in anisotropic functionclasses and application to adaptive estimation of evolutionary spectra. Ann. Statist.,25(1):38–76.Nussbaum, M. (1985). Spline smoothing in regression models and asymptotic efficiencyin L 2 . Ann. Statist., 13(3):984–997.Nussbaum, M. (1986). On the nonparametric estimation of regression functions that aresmooth in a domain in R k . Theory Probab. Appl., 31(1):118–125.Nussbaum, M. (1996). Asymptotic equivalence of density estimation and Gaussian whitenoise. Ann. Statist., 24(6):2399–2430.Pinsker, M. S. (1980). Optimal filtration of square-integrable signals in Gaussian noise.Problems of Information Transmission, 16:52–68.Piterbarg, V. I. (1996). Asymptotic methods in the theory of Gaussian processes and fields,volume 148 of Translations of Mathematical Monographs. American Mathematical Society,Providence, RI. Translated from the Russian by V. V. Piterbarg, Revised by theauthor.Stechkin, S. B. (1968). Best approximation of linear operators. Math. Notes, 1:91–99.Stein, C. M. (1981). Estimation of the mean of a multivariate normal distribution. Ann.Statist., 9(6):1135–1151.Stone, C. J. (1980). Optimal rates of convergence for nonparametric estimators. Ann.Statist., 8(6):1348–1360.Stone, C. J. (1982). Optimal global rates of convergence for nonparametric regression.Ann. Statist., 10(4):1040–1053.Stone, C. J. (1984). An asymptotically optimal window selection rule for kernel densityestimates. Ann. Statist., 12(4):1285–1297.Stone, C. J. (1985). Additive regression and other nonparametric models. Ann. Statist.,13(2):689–705.Stone, C. J. (1986). The dimensionality reduction principle for generalized additive models.Ann. Statist., 14(2):590–606.Tsybakov, A. B. (1997). Asymptotically efficient estimation of a signal in L 2 with generalloss functions. Problems of Information Transmission, 33(1):78–88.Tsybakov, A. B. (1998). Pointwise and sup-norm sharp adaptive estimation of functionson the Sobolev classes. Ann. Statist., 26(6):2420–2469.Tsybakov, A. B. (2003). Optimal rates of aggregation. In Computational Learning Theoryan Kernel Machines, Lectures Notes in Artificial Intelligence, pages 303–313. Springer,Heidelberg.
126 BibliographieTsybakov, A. B. (2004). Introduction à l’estimation non-paramétrique, volume 41 of Mathématiques& Applications (Berlin) [Mathematics & Applications]. Springer-Verlag,Berlin.Vainberg, M. M. (1964). Variational methods for the study of nonlinear operators. Witha chapter on Newton’s method by L. V. Kantorovich and G. P. Akilov. Translated andsupplemented by Amiel Feinstein. Holden-Day Inc., San Francisco, Calif.Vainberg, M. M. (1972). Variational method and method of monotone operators in thetheory of nonlinear equations. Izdat. “Nauka”, Moscow.Yang, Y. (2000). Mixing strategies for density estimation. Ann. Statist., 28(1):75–87.
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THÈSE DE DOCTORAT DE L’UNIVERSIT
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Table des matières 1Table des mati
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Table des matières 3.2 Un théorè
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1.1. Objet de la thèse 5blanc gaus
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1.1. Objet de la thèse 7calculées
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1.1. Objet de la thèse 9Un troisi
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1.2. Principaux résultats 11où β
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1.2. Principaux résultats 13le Cha
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1.2. Principaux résultats 15où K
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1.2. Principaux résultats 17de fon
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1.2. Principaux résultats 19L > 0
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Chapitre 2Asymptotically exact esti
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2.2. The main result and the estima
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2.2. The main result and the estima
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2.3. Proofs 27We study the stochast
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2.3. Proofs 29[for ε introduced in
Page 34 and 35:
2.3. Proofs 31where P θ,X is the d
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2.3. Proofs 33as n → ∞ and usin
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2.3. Proofs 35Thus some algebra and
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2.3. Proofs 37Such choice of n is p
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2.4. Appendix of Chapter 2 39• Fi
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3.1. Introduction 41where ψ n = (
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3.2. The estimator and main result
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3.3. Upper bound 45Proof. The stoch
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3.4. Lower bound 473.4 Lower boundB
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3.4. Lower bound 49With these preli
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3.5. Appendix of Chapter 3 51• If
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3.5. Appendix of Chapter 3 53where
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Chapitre 4Asymptotically exact mini
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4.2. Main result 57with ˜β define
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4.2. Main result 59(cf. Chapter 5 f
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4.3. Upper bound 61Then we have( )
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4.3. Upper bound 63where N Bj (u) i
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4.4. Lower bound 65is a Gaussian ra
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5.1. Cadre général de l’optimal
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5.2. Application au problème d’a
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Page 78 and 79: 5.2. Application au problème d’a
Page 84 and 85: 5.3. Lien entre l’optimal recover
Page 90 and 91: 5.4. Etudes des constantes 87sente
Page 92 and 93: 6.1. Introduction 89(where ‖g‖
Page 94 and 95: 6.2. Main results 91and we denote b
Page 96 and 97: 6.2. Main results 936.2.4 Exact asy
Page 98 and 99: 6.2. Main results 952. These result
Page 100 and 101: 6.3. Some preliminary results 97Pro
Page 102 and 103: 6.4. Proof of Theorem 6.1 99Let 0 <
Page 104 and 105: 6.4. Proof of Theorem 6.1 101where
Page 106 and 107: 6.5. Proof of Theorem 6.2 103withA
Page 108 and 109: 6.5. Proof of Theorem 6.2 105The qu
Page 110 and 111: 6.6. Proof of Theorem 6.3 107andwhe
Page 112 and 113: 6.6. Proof of Theorem 6.3 109By Pro
Page 114 and 115: 6.7. Proof of Theorem 6.4 111The pr
Page 116 and 117: 6.8. Proofs of the lemmas and propo
Page 122 and 123: 119Annexe.1 Résultats sur les proc
Page 124 and 125: Bibliographie 121BibliographieAdler
Page 126 and 127: Bibliographie 123Fuller, A. T. (196
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