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

6.2. Main results 936.2.4 Exact asymptotics for particular forms of the set BFor β ∈ B and j ∈ {1, 2, 3}, we consider() 1/22‖K β ‖ 2η j (β) =2σ 2 c j (β)n ∏ di=1 h i,j(β)(2β + 1) log n ,where c 3 (β) = c 1 (β). We select the vector ˆβ (p) ∈ B defined by{ˆβ (p) = max β ∈ B : ∀γ < β, ‖ ˆf β,1 − ˆf}γ,1 ‖ ∞ ≤ η 1 (γ)We have the following theorem for particular forms of the set B, which include the setsB of isotropic classes.Theorem 6.2. We suppose that the set B satisfies the property (P ):(P ) For all β = (β 1 , . . . , β d ), γ = (γ 1 , . . . , γ d ) ∈ B, if β ≤ γ then for all i = 1, . . . , dβ i < γ i .Then, the estimator ˜f (p) = ˆf ˆβ(p) ,1is asymptotically exact adaptive, i.e. it satisfies thecondition (6.3) with the constant C β defined in (6.6):lim inf R n (θ n ) = lim R n ( ˜f (p) ) = 1n→∞ θ nn→∞Remark. The estimator ˜f (p) is obtained using the method of Lepski. The constant C β hasthe form of the constant obtained by Lepski (1992) in the case d = 1 where we replace βby β.6.2.5 Upper bounds for anisotropic classesIf the set B does not satisfy Condition (P), Theorem 6.2 is not true anymore. In this subsection,Theorem 6.3 gives an upper bound for any finite set B ⊂ (0, 1] d and Theorem 6.4improves this upper bound when B contains only two vectors.We consider new estimators defined for β and γ ∈ B and t ∈ [0, 1] d by∫ˆf β∗γ,2 (t) = K β∗γ,2 (t − u)dY uwhereK β∗γ,2 = K β,2 ∗ K γ,2 .We consider the vector ˆβ ani ∈ B defined by{ˆβ ani = max β ∈ B : ∀γ < β, ‖ ˆf β∗γ,2 − ˆf}γ,2 ‖ ∞ ≤ η 2 (γ) .