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**

**Example text**

1, the fact that s1 belongs to L∞ is essentially a necessary condition to have a control on the L2 -risk of estimators of s1 . d. observations and the estimation of the intensity s = ns1 from a Poisson process. This suggests to adapt the known results from density estimation to intensity estimation for Poisson processes. We shall brieﬂy explain how it works, when the Poisson process X has an intensity s ∈ L∞ (λ) with L∞ -norm s ∞ . The starting point is to observe that, given an element ϕ ∈ L2 (λ), a natural N estimator of ϕ, s is ϕ(X) = ϕ dΛX = i=1 ϕ(Xi ).

1, the fact that s1 belongs to L∞ is essentially a necessary condition to have a control on the L2 -risk of estimators of s1 . d. observations and the estimation of the intensity s = ns1 from a Poisson process. This suggests to adapt the known results from density estimation to intensity estimation for Poisson processes. We shall brieﬂy explain how it works, when the Poisson process X has an intensity s ∈ L∞ (λ) with L∞ -norm s ∞ . The starting point is to observe that, given an element ϕ ∈ L2 (λ), a natural N estimator of ϕ, s is ϕ(X) = ϕ dΛX = i=1 ϕ(Xi ).

Let Φ(x) = sup{h(x) : h is convex and h(z) ≤ Φ(z) for all z ∈ [0, 1]}. Then sup |Φ(x) − Ψ(x)| ≤ sup |Φ(x) − Ψ(x)|. 0≤x≤1 0≤x≤1 Proof. Note that for all y ∈ [0, 1], either Φ(y) = Φ(y), or y is an interior point of a closed interval I over which Φ is linear. For such an interval, either supx∈I |Φ(x) − A Kiefer–Wolfowitz theorem 27 Ψ(x)| is attained at an endpoint of I (where Φ = Φ), or it is attained at an interior point, where Ψ < Φ. Since Φ ≤ Φ on [0, 1], it follows that sup |Φ(x) − Ψ(x)| ≤ sup |Φ(x) − Ψ(x)|.