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

6.2. Main results 91and we denote by B the set:B ={β (1) , . . . , β (l)} .We suppose that β (i) ≠ β (j) for all i, j ∈ {1, . . . , l} such that i ≠ j. As a consequence,a β ∈ B is matched to a unique β ∈ B via the relations (6.4) and (6.5). We define thefollowing relation of order in B: the vectors β and γ satisfyβ ≤ γ if and only if β ≤ γ.This is a relation of total order in B, thus the notion of maximum and minimum in B arewell-defined. We denote β max = max{β (i) , i = 1, · · · , l} (respectively β min = min{β (i) , i =1, · · · , l}) and β max (respectively β min ) the associated real number in B via (6.5).Remark. In the isotropic setting (i.e. for all β = (β 1 , . . . , β d ) ∈ B, β 1 = · · · = β d ), thisorder is the order β = (β 1 , . . . , β d ) ≤ γ = (γ 1 , . . . , γ d ) if and only if β 1 ≤ γ 1 .6.2.2 Three families of estimatorsHere we define three families of kernel estimators ( ˆf β,1 ) β∈B , ( ˆf β,2 ) β∈B and ( ˆf β,3 ) β∈B . Theyare close to the asymptotically exact estimator of Chapter 3, but the kernel is somewhatdifferent at the boundary. This is a consequence of the fact that the observations hereare for t ∈ R d whereas they were for t ∈ [0, 1] d in Chapter 3. The estimators are definedin the following way. For β ∈ B and j ∈ {1, 2, 3},∫ˆf β,j (t) = K β,j (t − u) dY u ,R ddefined for t = (t 1 , . . . , t d ) ∈ [0, 1] d , where for u = (u 1 , . . . , u d ) ∈ R dK β,j (u) =1h 1,j (β) · · · h d,j (β) K β(u1h 1,j (β) , . . . ,)u d.h d,j (β)For j ∈ {1, 2, 3}, the bandwidth h j = (h 1,j (β), . . . , h d,j (β)) satisfies for i ∈ {1, . . . , d}h i,j (β) =(C β λ j (β)L i( ) ) 1/βi β/(2β+1) log n,nwhere⎧1 for j = 1,⎪⎨ ( )λ j (β) = 2 − 2ββc2 (β) 2β+12β+1c 1for j = 2,(β)⎪⎩2 − 2β2β+1for j = 3,⎛ () ⎞ 1β 2β+1C β = ⎝σ 2β c 1 (β)(β + 1)L ∗ (β)⎠α(β)β 3, (6.6)