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

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6.2. Main results 91and we denote by B the set:B ={β (1) , . . . , β (l)} .We suppose that β (i) ≠ β (j) for all i, j ∈ {1, . . . , l} such that i ≠ j. As a consequence,a β ∈ B is matched to a unique β ∈ B via the relations (6.4) and (6.5). We define thefollowing relation of order in B: the vectors β and γ satisfyβ ≤ γ if and only if β ≤ γ.This is a relation of total order in B, thus the notion of maximum and minimum in B arewell-defined. We denote β max = max{β (i) , i = 1, · · · , l} (respectively β min = min{β (i) , i =1, · · · , l}) and β max (respectively β min ) the associated real number in B via (6.5).Remark. In the isotropic setting (i.e. for all β = (β 1 , . . . , β d ) ∈ B, β 1 = · · · = β d ), thisorder is the order β = (β 1 , . . . , β d ) ≤ γ = (γ 1 , . . . , γ d ) if and only if β 1 ≤ γ 1 .6.2.2 Three families of estimatorsHere we define three families of kernel estimators ( ˆf β,1 ) β∈B , ( ˆf β,2 ) β∈B and ( ˆf β,3 ) β∈B . Theyare close to the asymptotically exact estimator of Chapter 3, but the kernel is somewhatdifferent at the boundary. This is a consequence of the fact that the observations hereare for t ∈ R d whereas they were for t ∈ [0, 1] d in Chapter 3. The estimators are definedin the following way. For β ∈ B and j ∈ {1, 2, 3},∫ˆf β,j (t) = K β,j (t − u) dY u ,R ddefined for t = (t 1 , . . . , t d ) ∈ [0, 1] d , where for u = (u 1 , . . . , u d ) ∈ R dK β,j (u) =1h 1,j (β) · · · h d,j (β) K β(u1h 1,j (β) , . . . ,)u d.h d,j (β)For j ∈ {1, 2, 3}, the bandwidth h j = (h 1,j (β), . . . , h d,j (β)) satisfies for i ∈ {1, . . . , d}h i,j (β) =(C β λ j (β)L i( ) ) 1/βi β/(2β+1) log n,nwhere⎧1 for j = 1,⎪⎨ ( )λ j (β) = 2 − 2ββc2 (β) 2β+12β+1c 1for j = 2,(β)⎪⎩2 − 2β2β+1for j = 3,⎛ () ⎞ 1β 2β+1C β = ⎝σ 2β c 1 (β)(β + 1)L ∗ (β)⎠α(β)β 3, (6.6)
92 Sharp adaptive estimation in sup-norm for d-dimensional Hölder classesand( λµ(β) = maxλ>β 2λ + 1 −c 1 (β) = 1 + p(β max − β)2β max + 1 ,c 2 (β) = 1 + pµ(β)(2β + 1){ γminγ 0. We have the following lower boundwhich is valid for any finite set B satisfying the conditions of Subsection 6.2.1.Theorem 6.1. The minimax risk in sup-norm on B satisfieslim infn→∞infθ nR n (θ n ) ≥ 1.
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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
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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
Page 44 and 45: 3.1. Introduction 41where ψ n = (
Page 46 and 47: 3.2. The estimator and main result
Page 48 and 49: 3.3. Upper bound 45Proof. The stoch
Page 50 and 51: 3.4. Lower bound 473.4 Lower boundB
Page 52 and 53: 3.4. Lower bound 49With these preli
Page 54 and 55: 3.5. Appendix of Chapter 3 51• If
Page 56 and 57: 3.5. Appendix of Chapter 3 53where
Page 58 and 59: Chapitre 4Asymptotically exact mini
Page 60 and 61: 4.2. Main result 57with ˜β define
Page 62 and 63: 4.2. Main result 59(cf. Chapter 5 f
Page 64 and 65: 4.3. Upper bound 61Then we have( )
Page 66 and 67: 4.3. Upper bound 63where N Bj (u) i
Page 68 and 69: 4.4. Lower bound 65is a Gaussian ra
Page 70 and 71: 5.1. Cadre général de l’optimal
Page 72 and 73: 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 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
Page 128 and 129: Bibliographie 125Micchelli, C. A. a
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THÈSE DE DOCTORAT DE L'UNIVERSITÉ PARIS 6 Spécialité ...

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