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By Eric A. Cator, Cor Kraaikamp, Hendrik P. Lopuhaa, Jon A. Wellner, Geurt Jongbloed

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Additional info for Asymptotics: particles, processes and inverse problems. Festschrift for Piet Groeneboom

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These estimators span a D-dimensional linear subspace of L2 (λ) with an orthonormal basis ϕ1 , . . , ϕD , D ≤ |M|. Working conditionally with respect to X1 , we use X2 to build a projection estimator s˜(X2 ) of (1 − p)s belonging to the linear span of the estimators sˆm (X1 ). 1 extends straightforwardly to Poisson processes to give Theorem 2. 3)   2 Es s˜(X) − (1 − p)s 2 2 ps − ≤ Es  inf θ∈RM θm sˆm (X1 ) m∈M 2 Setting sˆ(X) = s˜(X)/(1 − p) leads to Es sˆ(X) − s 2 2 ≤ 1 inf Es (1 − p)2 m∈M ps − sˆm (X1 ) 2 2  + (1 − p) s + ∞ |M|.

K. • Expanding ∆j+1 a around ∆j a: We have ∆j+1 a = aj+1 − aj = aj − aj−1 + [aj+1 − aj − (aj − aj−1 )] aj+1 − aj = ∆j a + ∆j a − 1 = ∆j a + ∆j a ǫ3,j aj − aj−1 A Kiefer–Wolfowitz theorem 21 where ǫ3,j = = f (a∗j ) f (a∗j ) − f (a∗j+1 ) aj+1 − aj −1= − 1 = aj − aj−1 f (a∗j+1 ) f (a∗j+1 ) −f ′ (a∗∗ j ) ∗ (a − a∗j ). ∗ f (aj+1 ) j+1 Thus, |ǫ3,j | ≤ = |f ′ (aj−1 )| (aj+1 − aj−1 ) f (aj+1 ) |f ′ (aj−1 )| f (aj+1 ) 1 1 + kn f (a∗j ) kn f (a∗j+1 ) 2 |f ′ (aj−1 )| 1 |f ′ (aj−1 )| f (aj−1 ) 1 =2 2 2 f (aj+1 ) kn f (aj−1 ) f (aj+1 ) kn 1 |f ′ (aj−1 )| 1 |f ′ (aj−1 )| 1 = 32 2 ≤ 32˜ γ1 .

1. The basics of Poisson processes Let us denote by Q+ (X ) the cone of finite positive measures on the measurable space (X , E). Given an element µ ∈ Q+ (X ), a Poisson process on X with mean measure µ is a point process X = {X1 , . . d. with distribution µ1 = µ/µ(X ). Equivalently, the Poisson process can be viewed as N a random measure ΛX = i=1 δXi , δx denoting the Dirac measure concentrated at the point x. Then, whatever the partition A1 , . . fr AMS 2000 subject classifications: Primary 62M30, 62G05; secondary 62G10, 41A45, 41A46.

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