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

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3.4. Lower bound 49With these preliminaries, we can now prove inequality (3.7). For any f ∈ Σ(β, L) andfor any estimator ̂f n , using the monotonicity of w and the Markov inequality, we obtainthat(E f[w ψn −1 ‖ ̂f)][n − f‖ ∞ ≥w(C 0 (1 − ε))P f ψn −1 ‖ ̂f]n − f‖ ∞ ≥ C 0 (1 − ε) .Since Σ ′ ⊂ Σ(β, L), it is enough to prove that lim n→∞ Λ n = 1, where[Λ n = inf sup P f ψn −1 ‖ ̂f]n − f‖ ∞ ≥ C 0 (1 − ε) .̂f n f∈Σ ′We have max j=1,...,M | ̂f n (a j ) − f(a j )| ≤ ‖ ̂f n − f‖ ∞ . Setting ˆθ j = ̂f n (a j )C 0 ψ n and usingthe fact that f(a j , θ) = C 0 ψ n θ j for θ ∈ [−1, 1] M , we see thatwhere C n =Λ n ≥ inf sup P θ (C n ),ˆθ∈R M θ∈[−1,1] M{}max j=1,...,M |ˆθ j − θ j | ≥ 1 − ε and ˆθ = (ˆθ 1 , . . . , ˆθ M ) ∈ R Mwith respect to y = {Y t , t ∈ [0, 1] d }. We have∫Λ n ≥ infP θ (C n )π(dθ),ˆθ∈R M {−(1−ε),1−ε} Mis measurablewhere π is the prior distribution on θ, π(dθ) = ∏ Mj=1 π j(dθ j ), where π j is the Bernoullidistribution on {−(1 − ε), 1 − ε} that assigns probability 1/2 to −(1 − ε) and to (1 − ε).Since P θ is absolutely continuous with respect to P 0 (see Proposition 3.3), we have∫Λ n ≥ infˆθ∈R M∫= infˆθ∈R M( )dP θE 0 I Cn π(dθ)dP 0( M)∏ ϕ vn (y j − θ j )E 0 I Cnπ(dθ).ϕ vn (y j )By the Fubini and Fatou theorems, we can write∫( ∫ M)1 ∏Λ n ≥1 − sup ∏ Mˆθ∈R M j=1 ϕ I {|θj −ˆθv n(y j )j |
50 Exact minimax estimation in sup-norm for anisotropic Hölder classesadmits the solution ˜θ j (y j ) = (1 − ε)I {yj ≥0} − (1 − ε)I {yj
Page 1: THÈSE DE DOCTORAT DE L’UNIVERSIT
Page 4 and 5: Table des matières 1Table des mati
Page 6 and 7: Table des matières 3.2 Un théorè
Page 8 and 9: 1.1. Objet de la thèse 5blanc gaus
Page 10 and 11: 1.1. Objet de la thèse 7calculées
Page 12 and 13: 1.1. Objet de la thèse 9Un troisi
Page 14 and 15: 1.2. Principaux résultats 11où β
Page 16 and 17: 1.2. Principaux résultats 13le Cha
Page 18 and 19: 1.2. Principaux résultats 15où K
Page 20 and 21: 1.2. Principaux résultats 17de fon
Page 22 and 23: 1.2. Principaux résultats 19L > 0
Page 24 and 25: Chapitre 2Asymptotically exact esti
Page 26 and 27: 2.2. The main result and the estima
Page 28 and 29: 2.2. The main result and the estima
Page 30 and 31: 2.3. Proofs 27We study the stochast
Page 32 and 33: 2.3. Proofs 29[for ε introduced in
Page 34 and 35: 2.3. Proofs 31where P θ,X is the d
Page 36 and 37: 2.3. Proofs 33as n → ∞ and usin
Page 38 and 39: 2.3. Proofs 35Thus some algebra and
Page 40 and 41: 2.3. Proofs 37Such choice of n is p
Page 42 and 43: 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 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 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 118 and 119:
Page 120 and 121:
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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