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Asymptotics: particles, processes, and inverse problems: by Eric A. Cator, Cor Kraaikamp, Hendrik P. Lopuhaa, Jon A.

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

Cator E.A., et al. (eds.) Asymptotics.. debris, strategies and inverse difficulties (Inst.Math.Stat., 2007)(ISBN 0940600714)-o

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1214/074921707000000265 Model selection for Poisson processes Lucien Birg´ e1 Universit´ e Paris VI Abstract: Our purpose in this paper is to apply the general methodology for model selection based on T-estimators developed in Birg´e [Ann. Inst. H. Poincar´ e Probab. Statist. 42 (2006) 273–325] to the particular situation of the estimation of the unknown mean measure of a Poisson process. We introduce a Hellinger type distance between finite positive measures to serve as our loss function and we build suitable tests between balls (with respect to this distance) in the set of mean measures.

3. Intensity estimation A case of particular interest occurs when we have at hand a reference positive measure λ on X and we assume that µ λ with dµ/dλ = s, in which case s is called the intensity (with respect to λ) of the process with mean measure µ. + Denoting by L+ i (λ) the positive part of Li (λ) for i = 1, 2, we observe that s ∈ L1 (λ), √ + + s ∈ L2 (λ) and µ ∈ Qλ = {µt = t · λ, t ∈ L1 (λ)}. 10) √ H(t, u) = H(µt , µu ) = 1/ 2 √ √ t− u 2 for t, u ∈ L+ 1 (λ), where · 2 stands for the norm in L2 (λ).

And Woodroofe, M. (2007). A Kiefer–Wolfowitz comparison theorem for Wicksell’s problem. Ann. Statist. 35. To appear. [26] Wang, Y. (1994). The limit distribution of the concave majorant of an empirical distribution function. Statist. Probab. Lett. 20 81–84. MR1294808 IMS Lecture Notes–Monograph Series Asymptotic: Particles, Processes and Inverse Problems Vol. 1214/074921707000000265 Model selection for Poisson processes Lucien Birg´ e1 Universit´ e Paris VI Abstract: Our purpose in this paper is to apply the general methodology for model selection based on T-estimators developed in Birg´e [Ann.

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