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

118 Sharp adaptive estimation in sup-norm for d-dimensional Hölder classeswhere κ n is of order ψ n (β)/η 2 (β ′ ),andUsing Proposition 6.2, sinceP 3 (n) = P f [‖Z β∗β ′ ,2‖ ∞ > ψ n ((β ′ + β)/2)(1 + κ n )] ,P 4 (n) = P f[‖Z β ′ ,2‖ ∞ > η 2 (β ′ ) (1 + κ n )η 2 2(β ′ ) ∏ di=1 h i,2(β ′ )2‖K β ′‖ 2 2σ 2 =we obtain that for n large enough(1 − ψ n( β′ +βη 2 (β ′ )2)()12β ′ + 1 + pµ(β′ ) log n,P 4 (n) ≤ D 11 (log n) − 12β ′ +1 exp {− (pµ(β ′ )) log n} . (6.56)Using Lemma 6.4, it can be proved that P 3 (n) is negligible with respect to P 4 (n) asn → ∞. The relation (6.56) implies the lemma.Proof of Lemma 6.10)].Let f ∈ Σ(γ, L). By Lemma 6.3, we have thatThenwhereand‖b β∗γ,1 (·, f) − b β,1 (·, f)‖ ∞ ≤ ψ n(γ)λ 3 (γ).2γ + 1[P f ‖ ˆf β∗γ,1 − ˆf β,1 ‖ ∞ ≥ ψ ]n(γ)λ 3 (γ)(1 + ρ n ) ≤ P 5 (n) + P 6 (n),2γ + 1[P 5 (n) = P f ‖Z β∗γ,1 ‖ ∞ > ψ ]n(γ)λ 3 (γ)ρ n,2(2γ + 1)[P 6 (n) = P f ‖Z β,1 ‖ ∞ > ψ ]n(γ)λ 3 (γ)ρ n.2(2γ + 1)Using Proposition 6.2, since ρ n = ψ n((β+γ)/2)ψ n, we obtain that(γ){} ⎧√ ⎫P 6 (n) ≤ D 1˜h 1 (β) exp − λ2 3(γ)ψn((β 2 + γ)/2)n˜h 1 (β)⎨exp8‖K β ‖ 2 2σ 2 (2γ + 1) 2 ⎩ −D 2λ 3 (γ)ψ n ((β + γ)/2) n˜h 1 (β) ⎬√2(2γ + 1) log ˜h ⎭ .1 (β)(6.57)Using Lemma 6.4, we have that P 5 (n) satisfies the same inequality as (6.57) but withdifferent constants D 1 and D 2 . Now, since β > γ, we havewith D 17 ∈ R, which includes the lemma.λ 2 3(γ)ψ 2 n((β + γ)/2)n˜h 1 (β)8‖K β ‖ 2 2σ 2 (2γ + 1) 2 = D 16 (log n) D 17n D 18,