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

34 Exact estimation in sup-norm for nonparametric regression with random designBy Bernstein’s inequality applied to the variables Z i , we have()⎛P X | 1 n∑n(δ n − |δ k,h |)Z i | < δ n − |δ k,h | ≥ 1 − 2 exp ⎝− ( 2ni=12 K 2 max µ 1h+ 2(δ n−|δ k,h |)K max3h⎞) ⎠ .Using the fact that δ k,h = O(h), we obtain that for n large enough, there exists a constantc A independent of k such thatP X (A(k)) ≥ 1 − 2 exp ( −c A nhδ 2 n).From this we deduce easily the result about A n because card{k} ≤ n m .Proof of Proposition 2.1Let f ∈ Σ(β, L, Q). We have[ ( ) ] √√E f w ψ−1n ‖f − θn‖ ∗ ∞ IB C n ≤ E f [w 2 (ψn −1 ‖f − θn‖ ∗ ∞ )] P f (Bn C )√≤ E f (1 + (ψn −1 ‖f − θn‖ ∗ ∞ ) γ ) 2√ P X (Bn C )since the event B n only depends on X,≤ √ √(2 1 + E f (ψ−1n ‖f − θn‖ ∗ ∞ ) 2γ) √PX(Bn C ).Now E f((ψ n−1 ‖f − θn‖ ∗ ∞ ) 2γ) ≤ ψn −2γ D 6 (Q 2γ + E f ‖θn‖ ∗ 2γ ∞). Some algebra and the factthat(max1nhn∑(Xj − x kKhj=1), δ n)≥ δ n ,yield E f ‖θ ∗ n‖ 2γ ∞ = O(n γ 1), with some γ 1 ≥ 0. From the relations above and Lemma 2.1,we deduce that lim n→∞ E f[w (ψ−1n ‖f − θ ∗ n‖ ∞ ) I B C n]= 0.Proof of Proposition 2.2Let f ∈ Σ(β, L, Q) and x k ∈ [h, 1−h]. Consider n large enough such that δ n ≤ µ(x k )−δ n .We have on B nµ(x k ) − δ n ≤ 1 n∑( )Xj − x kK.nhhj=1