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

ISBN-10: 0940600714

ISBN-13: 9780940600713

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

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

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The fact that the errors of likelihood ratio tests between two probabilities are controlled by their Hellinger affinity justifies the introduction of the Hellinger distance as the natural loss function for density estimation, as shown by Le Cam [26]. It also motivates the choice of H q as a natural loss function for estimating the mean measure of a Poisson process. For simplicity, we shall first µ(X), µ)]. 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.

2007). Marshall’s lemma [7] Du for convex density estimation. In Asymptotics, Particles, Processes, and Inverse Problems, IMS Lecture Notes Monogr. Ser. Inst. Math. , Beachwood, OH. To appear. [8] Durot, C. -S. (2003). On the distance between the empirical process and its concave majorant in a monotone regression framework. Ann. Inst. H. Poincar´e Probab. Statist. 39 217–240. MR1962134 [9] Groeneboom, P. and Jongbloed, G. (1995). Isotonic estimation and rates of convergence in Wicksell’s problem.

We also provide an illustration based on nonlinear approximating models. Section 4 is devoted to various applications of our method based on families of linear models. This section essentially relies on results Model selection for Poisson processes 37 from approximation theory about the approximation of different classes of functions (typically smoothness classes) by finite dimensional linear spaces in L2 . We also indicate how to mix different families of models and introduce an asymptotic point of view which allows to consider convergence rates and to make a parallel with density estimation.

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Asymptotics: particles, processes, and inverse problems: festschrift for Piet Groeneboom by Eric A. Cator, Cor Kraaikamp, Hendrik P. Lopuhaa, Jon A. Wellner, Geurt Jongbloed

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