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

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2.2. The main result and the estimator 23and that the exact constant depends on f β (0). For 0 < β ≤ 1, the solution of (2.6)is known (see Korostelev (1993) or Donoho (1994a)) and the kernel used by Korostelevdefined in (2.5) is equal to that defined in (2.7) up to a renormalization on the support.However the function f β is not known for β > 1, except for β = 2. Korostelev andNussbaum (1999) have found the exact constant and asymptotically exact estimator forthe density model in sup-norm. Lepski (1992) has studied the exact constant in the caseof adaptation for the white noise model. The sup-norm estimation is only one of theapproaches studied in the nonparametric literature. For the L 2 -norm risk, one can findoverview of results on exact minimax and adaptive estimation in the books of Efromovich(1999) and Tsybakov (2004)Our results are the following. In Section 2.2, we give an asymptotically exactestimator θn ∗ and the exact constant for the regression model with random design. Ifthe density µ is uniform (µ 0 = 1), then the constant is equal to w(C 0 ) (the constant ofKorostelev (1993)). As it could be expected, the exact constant and the asymptoticallyexact estimator θn ∗ depend on the minimum value of the design density µ 0 . It means thatthe asymptotically minimax estimators contribute to the sup-norm risk essentially at thepoints where we have less observations. The estimator θn ∗ that is proposed in Section 2.2is close to a Nadaraya-Watson estimator and is independent of Q. The proofs are givenin Section 2.3.2.2 The main result and the estimatorIn this section, we define an estimator θ ∗ n. We shall prove in Subsection 2.3.1 that θ ∗ n is anasymptotically exact estimator. This estimator is close to a Nadaraya-Watson estimatorwith the kernel K defined in (2.5). The bandwidth of θ ∗ n ish =( ) 1C′0 ψβ n ,LwithC ′ 0 =(σ 2β L( ) ) 1β 2β+1β + 1.2β 2 µ 0First let us define θ ∗ n in a regular grid of points x k = km n∈ [0, 1] for k ∈ {1, . . . , [ n ]}, withmm = [δ n nψ 1 βn + 1], δ n = 1 and [x] denotes the integer part of x. To account for thelog nboundary effects, we need to introduce other kernels:K 1 (t) = 2K(t)I [0,1] (t), K 2 (t) = 2K(t)I [−1,0] (t) for t ∈ R.
24 Exact estimation in sup-norm for nonparametric regression with random designThe estimator θn ∗ is defined for k ∈ {1, . . . , [ n ]} bymθ ∗ n(x k ) =1nh(max∑ nj=1 K (Xj −x kh1nh)∑ nj=1 K (Xj −x khY j), δ n), (2.8)if x k ∈ [h, 1−h]. If x k ∈ [0, h) (respectively x k ∈ (1−h, 1]), θn(x ∗ k ) is defined by (2.8) whereK is replaced by K 1 (respectively by K 2 ). Finally the function θn ∗ is defined to be thepolygonal function connecting the points (x k , θn(x ∗ k )) for k ∈ {1, . . . , [ n ]}. Moreover, wemput θn(x) ∗ = θn(x ∗ 1 ) if x ∈ [0, x 1 ] and if x [nm ] < 1 we put θn(x) ∗ = θn(x ∗ [nm ] ) for x ∈ [x [nm ] , 1].The results we obtain are the following:Theorem 2.1. We consider the model and the assumptions defined in Section 2.1. Wesuppose that the function f ∈ Σ(β, L, Q), with 0 < β ≤ 1. The estimator θ ∗ n satisfieslim inf R n (θ n ) = lim R n (θ ∗n→∞ θ nn→∞n) = w(C 0).′We are going to prove Theorem 2.1 in two steps: the upper bound (Subsection 2.3.1)and the lower bound (Subsection 2.3.2). Let 0 < ε < 1/2. In Subsection 2.3.1, we showthat θn ∗ satisfieslim supn→∞supf∈Σ(β,L,Q)In Subsection 2.3.2, we prove thatlim infn→∞infθ n[ ( )] (E f w ‖θ∗n − f‖ ∞ ψn−1 ≤ w C′0 (1 + ε) 2) . (2.9)sup E f [w(‖θ n − f‖ ∞ ψn −1 )] ≥ w(C 0(1 ′ − ε)). (2.10)f∈Σ(β,L,Q)Since 0 < ε < 1/2 in (2.9) and (2.10) can be arbitrarily small and w is a continuousfunction, this proves Theorem 2.1.Remarks:∑ ( )(i) We introduce the cut-off δ n in (2.8) to account for the case 1 nnh j=1 K Xj −x k= 0h∑ ( )1which leads to a zero denominator. As it is proved in Lemma 2.1, nnh j=1 K Xj −x k−h∑ ( )µ(x k ) tends to 0 in probability as soon as n tends to ∞, so that 1 nnh j=1 K Xj −x k= 0hessentially does not occur.(ii) The estimator θn ∗ does not depend on Q, but it depends on µ 0 . It is possible toconstruct an asymptotically exact estimator independent of µ 0 and Q but the proof israther technical. For this purpose, we cut the sample (X 1 , . . . , X n ) in two parts of sizeα n and n − α n , where α n is an integer such that α n → ∞ and α n /n → 0 as n → ∞. Weestimate µ 0 with the part (X 1 , . . . , X αn ) of the sample by( ) kˆµ 0 = min ˆµ n ,k=1,...,n n
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 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 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 74 and 75: 5.2. Application au problème d’a
Page 76 and 77:
5.2. Application au problème d’a
Page 78 and 79:
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
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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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