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

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2.2. The main result and the estimator 25∑ ( )where ˆµ n (x) = 1 αnα ng n i=1 K X i −xg nand g n such that g n → 0 and α n g n → ∞. We constructan estimator of f in the same way as θn ∗ except we only use the part (X αn+1, . . . , X n )of the sample and we replace µ 0 by ˆµ 0 , if the latter is not zero, in C 0 ′ and h. The resultsare similar to those of this paper for this estimator but one needs to consider the lawconditioned by (X 1 , . . . , X αn ).(iii) We have only solved the problem of exact constant and asymptotically exact estimatorfor the Hölder classes Σ(β, L, Q) such that 0 < β ≤ 1. In this case we have an explicitform for the constant and the estimator. An extension to β > 1 is possible but it doesnot give realizable estimators (since the solution f β of the problem (2.6) is not explicitlyknown except for β = 2). A similar result could be found and the exact constant will beC 1 with( (C 1 = f β (0) σ 2β Lθ ∗ n(t) =2µ 0 (2β + 1)) β) 12β+1.The analogue of inequality (2.9) for β > 1 holds for example for the estimator θn ∗ definedfor t ∈ [0, 1] by:∑ ( )1 nnh j=1 K Xj −tYh j(max1nh∑ nj=1 K (Xj −th), δ n), (2.11)with certain modifications near the boundaries. To prove inequality (2.9) with this newestimator, we will use methods similar to those of Lepski and Tsybakov (2000), basedon the supremum of Gaussian processes. For β > 1, the proof of inequality (2.10) is thesame as that of Subsection 2.3.2, but we need to use the function f β and the fact thatf β is compactly supported. This was proved by Leonov (1997)He also proved that f β iscontinuous and even for all β > 1.(iv) Our result can be presumably extended to the white noise modeldY (t) = f(t)dt + σ(t)dW (t), t ∈ [0, 1],where W is a standard Wiener process and σ −2 serves to replace the density of the designpoints. In this model, the maximum of σ 2 corresponds to the minimum value of the designdensity µ 0 . An asymptotically exact estimator will be of the formθn(t) ∗ = 1 ∫ ( ) u − tK dY (u),h hwhere h a bandwidth that depends on n and K is defined in (2.7).(v) The constants L and β are supposed to be known, but using the techniques similar toLepski (1992), one can presumably obtain adaptive asymptotically exact estimator. Oneshould note however that the exact constant for adaptive estimator would be in generaldifferent.
26 Exact estimation in sup-norm for nonparametric regression with random design2.3 Proofs2.3.1 Proof of inequality (2.9)We define the event A n as{A n =maxx k ∈[h,1−h]∣∣µ(x k ) − 1nhn∑( ) }Xj − x ∣∣∣kK< δn .hj=1Similarly we define A 1,n (respectively A 2,n ) which are obtained by replacing K by K 1(respectively by K 2 ) and taking the supremum over x k ∈ [0, h) (respectively over x k ∈(1 − h, 1]). We define alsoA ′ n ={maxx k ∈[h,1−h]∣ µ(x k)(β + 1)2β + 1− 1nhn∑j=1( ) }K 2 Xj − x ∣∣∣k< δn ,hand the events A ′ 1,n (respectively A ′ 2,n) obtained by replacing K by K 1 (respectively byK 2 ) and taking the supremum over x k ∈ [0, h) (respectively over x k ∈ (1 − h, 1]). LetB n = A n ∩ A 1,n ∩ A 2,n ∩ A ′ n ∩ A ′ 1,n ∩ A ′ 2,n. We have the following result.Lemma 2.1. There exists c > 0 such thatP X (B n ) ≥ 1 − 12 n m exp ( −cnhδ 2 n),for n large enough, where P X is the joint distribution of X = (X 1 , . . . , X n ).The proof of the lemma is given in Subsection 2.3.3.Before proving inequality (2.9), we give four propositions studying the behaviour of∆ n = ψn−1 ‖f − θn‖ ∗ ∞ on B n and Bn C . We postpone their proofs to Subsection 2.3.3. Wedenote I B the indicator function of a set B which takes the value 1 on B and 0 otherwise.In the sequel, D 0 , D 1 ,. . . are positive constants.Proposition 2.1. We havelimsupn→∞ f∈Σ(β,L,Q)E f[w(ψ−1n ‖f − θ ∗ n‖ ∞)IB C n]= 0.Define the bias and the stochastic terms for x ∈ [0, 1]Proposition 2.2. The bias term satisfiesb n (x, f) = E f (θ ∗ n(x)I Bn ) − f(x)P X (B n ),Z n (x, f) = θ ∗ n(x)P X (B n ) − E f (θ ∗ n(x)I Bn ).lim supn→∞supf∈Σ(β,L,Q)ψ −1n ‖b n (., f)‖ ∞ ≤ C′ 02β + 1 .
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 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 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 78 and 79:
5.2. Application au problème d’a
Page 80 and 81:
Page 82 and 83:
Page 84 and 85:
5.3. Lien entre l’optimal recover
Page 86 and 87:
Page 88 and 89:
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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