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

62 Asymptotically exact minimax estimation in sup-norm for additive models(t 1 , . . . , t d ) ∈ R d , we have∫ ( d∑ρ 2 (s, t) =h(n)R d j=1( d∏≤D 2 h(n)j=1( ) ( )}1 uj − t j{K˜β) 2uj − s j− K˜βduh j h j h jh j) ( d∑j=11h j∣ ∣∣∣ t j − s jh j∣ ∣∣∣˜β) 2,where the last line is obtained first by integration over u ∈ R d and then by using theproperties of the function x ↦→ x˜β.Here and in what follows we denote by D j , j = 2, 3, . . .positive constants. Thereforewith d 1 (n) =√D 2 h(n) ∏ dj=1 h j.ρ(s, t) ≤ d 1 (n)d∑t j − s j∣ ∣j=1h 1+1/˜βj˜β, (4.10)Now for λ > 0, let N 1 (λ, S) be the minimal number of hyperrectangles with edges( ) 1/˜β ( 1/˜βof length h 1+1/˜β λ1+1/˜β∏1 d 1 (n)d , . . . , hλd d 1 (n)d)that cover a set S =di=1 S i where S iis an interval of R of length T i > 0. Now we choose λ = | log h| −1/2 and we suppose nlarge enough such that h < 1. We set N(h) = N 1 (λ, [0, 1] d ). Denote B 1 ,. . . ,B N(h) suchhyperrectangles that cover [0, 1] d . We haveN 1 (λ, S) ≤([d∏i=1T ih 1+1/˜βi(d1 (n)dλ) 1/˜β]+ 1where [x] denotes the integer part of the real x. We have for b ≥ 0P [‖ξ t ‖ ∞ ≥ b] ≤N(h)∑j=1P[sup |ξ t | ≥ bt∈B j]),. (4.11)Let j ∈ {1, . . . , N(h)}. Using Corollary 14.2 of Lifshits (1995) (cf. Annexe .1), we get forb ≥ 4 √ 2D(B j , v/2)P[sup |ξ t | ≥ bt∈B j]≤ 2P[sup ξ t ≥ bt∈B j](≤ 2 exp − 1 (b − 4 √ ) ) 22D(B2v 2 j , v/2) , (4.12)whereD(B j , v/2) =∫ v/20(log NBj (u) ) 1/2du,