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

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Extra resources for Asymptotics: particles, processes and inverse problems. Festschrift for Piet Groeneboom

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A. (2001). A canonical process for estimation of convex functions: the “invelope” of integrated Brownian motion +t4 . Ann. Statist. 29 1620–1652. , Jongbloed, G. and Wellner, J. A. (2001). Estimation of a convex function: characterizations and asymptotic theory. Ann. Statist. 29 1653–1698. MR1891742 [12] Hall, C. A. (1968). On error bounds for spline interpolation. J. Approximation Theory 1 209–218. MR0239324 [13] Hall, C. A. and Meyer, W. W. (1976). Optimal error bounds for cubic spline interpolation.

D , 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|. s ∞ |M| . 2) Es sˆm (X1 ) − ps 2 2 ≤ inf t − ps t∈S m 2 2 +p s ∞ Dm = p2 inf t−s t∈S m 2 2 +p s Choosing p = 1/2, we conclude that Es sˆ(X) − s 2 2 ≤ inf m∈M inf t∈S m t−s 2 2 +2 s ∞ Dm +2 s ∞ |M|.

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. Keywords and phrases: adaptive estimation, aggregation, intensity estimation, model selection, Poisson processes, robust tests.

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