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

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6.4. Proof of Theorem 6.1 99Let 0 < ε < 1/2. Let β ∈ B \βmax . In the following several quantities depend on β,but we do not indicate this dependence to simplify the notations. We consider the set offunctions f j,β (·) defined, for j ∈ {0, · · · , M}, by{(f j,β = ψ n (β)(1 − ε) 1 − ∑ df 0,β = 0,∣∣i=1 ∣ t i−a j,ih i,1 (β)∣ β i), j = 1, . . . , M+where the a j = (a j,1 , . . . , a j,d ) form a grid of points in [0, 1] d . This grid is defined in thefollowing manner. For[]1m i =2h i,1 (2 1/β + 1) − 1with [x] the integer part of x and M = ∏ di=1 m i, we consider the points a(l 1 , . . . , l d ) ∈[0, 1] d for l i ∈ {1, . . . , m i } and i ∈ {1, . . . , d}, such that:a(l 1 , . . . , l d ) = 2(2 1 β + 1) (h1,1 l 1 , . . . , h d,1 l d ) .To simplify the notation, we denote these points a 1 , . . . , a M and each a j takes the form:a j = (a j,1 , . . . , a j,d ).The functions f j,β satisfy the following lemma which can be proved as in Chapter 3.Lemma 6.6.1- f j,β ∈ Σ(β, L),2- ‖f j,β ‖ 2 2 = σ2 c 1 (β) log n(1−ε) 2n(2β+1)3- the functions f j,β have disjoint support.Here we come back to the study of ∆ n . Let j ∈ {1, . . . , M} and T n an estimator. Wehave, since f j,β (a k ) = (1 − ε)ψ n (β)δ j,k ,ψ −1n(β)‖T n − f j,β ‖ ∞ ≥ψn−1 (β) max |T n(a k ) − f j,β (a k )|1≤k≤M ∣∣ ∣∣≥(1 − ε) max ∣ˆθ k − δ j,k ,1≤k≤Mwhere δ j,k is the Kronecker delta and ˆθ k = Tn(a k)ψn−11−ε(β)ψ −1n (β)‖T n − f j,β ‖ ∞ ≥ d(ˆθ, θ j ),. As a consequencewhere ˆθ = (ˆθ 1 , . . . , ˆθ M ), θ j = (δ 1,j , . . . , δ M,j ), and d(u, v) = (1 − ε) max 1≤k≤M |u k − v k | fortwo vectors u = (u 1 , . . . , u M ) and v = (v 1 , . . . , v M ) of R M .
100 Sharp adaptive estimation in sup-norm for d-dimensional Hölder classesIn the same way∆ n (β) ≥ infT nψ −1n0≤k≤M E k(β)‖T n − f 0,β ‖ ∞ ≥ψn−1 (β) max |T n(a k )|1≤k≤Mwhere θ 0 is the null-vector of R M . Then we have(max ‖Tn − f k,β ‖ p ∞ψn −p (β) )≥ infˆθ∈R d max{E 0((ψn (β)ψ n (β max )≥ψn−1 (β max ) max |T n(a k )|1≤k≤M≥ψ n(β)ψ n (β max ) d(ˆθ, θ 0 ),) pd p (ˆθ, θ 0 )), max1≤k≤M E k(d p (ˆθ, θ k )) } , (6.13)where E k is the expectation E k = E fk,β . We denote also P k = P fk,β .Here we apply Theorem 6 of Tsybakov (1998) to the relation (6.13). We consider theM + 1 parameters {θ 0 , . . . , θ M } ⊂ [0, 1] M , the family of probability measures {P θj = P j },the loss function w(x) = x p and the distance d previously defined. The parameters θ jsatisfy d(θ i , θ k ) ≥ 1 − ε for i, k ∈ {0, . . . , M} and i ≠ k. We are now going to prove thatthere exists α n , with 0 ≤ α n ≤ 1 and lim n→∞ α n = 0, such that( ) dP0QdQ ≥ τ n ≥ 1 − α n , (6.14)pβ − εc 1(β)2β+1∑where Q = 1 MM k=1 P k, τ n = n −ν , ν = pβ max−(1 − ε/2) and ε is chosen2β max +1 2β+1small enough to have ν > 0.Indeed, if we prove (6.14), Theorem 6 of Tsybakov implies that∆ n (β) ≥) p((1 − α n )τ n (1 − 2ε) p ψn (β)εψ n (β max )( ) p .(1 − 2ε) p + τψn (β)εn ψ n (β max )We have lim n→∞ τ n(ψn(β)εψ n(β max)) p= +∞. Thenlim infn→∞ ∆ n(β) ≥ (1 − 2ε) p .Since ε can be arbitrarily small, we obtain that lim inf n→∞ ∆ n (β) ≥ 1 and it proves thetheorem.Here we prove (6.14). In the same way as Tsybakov (1998), we have that under P i{dP k exp {ξ=k v n + vn}2 k = idP 0 exp {ξ k v n − vn} 2 k ≠ i,
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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
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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
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 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 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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