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

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Is not too large in a suitable sense to be explained below. We do not know of any other estimation method allowing to deal with model selection in such a generality and with as few assumptions. The main drawback of the method is its theoretical nature, eﬀective computation of the estimators being typically computationally too costly for permitting a practical implementation. In order to give a more precise idea of what this paper is about, we need to start by recalling a few well-known facts about Poisson processes that can, for instance, be found in Reiss [29].

In Statistical Decision Theory and Related Topics. II (Proc. , 1976) 193–211. Academic Press, New York. MR0443202 ¨, H. P. (2006). The limit process of the dif[16] Kulikov, V. N. and Lopuhaa ference between the empirical distribution function and its concave majorant. Statist. Probab. Lett. 76 1781–1786. MR2274141 [17] Marshall, A. W. (1970). Discussion on Barlow and van Zwet’s paper. In Nonparametric Techniques in Statistical Inference (M. L. ). Proceedings of the First International Symposium on Nonparametric Techniques held at Indiana University, June 1969 174–176.

This implies that S is a D-model with parameters η, (log K)/4 and 1. 2 √log K. 2k log(c( N/k + 1)), this inequality holds since η ≥ 2 k, hence K ≤ [c( N/k + 1)]k . The number of tests required for building the T-estimator is |S|(|S| − 1) < K 2 . For N of the order of 100 and k as small as 5, K 2 is of the order of 1010 . This toy example illustrates the diﬃculty of implementing the algorithm. More realistic ones would be much worse. 4. Applications with linear models We now assume that µ = µs = s · λ and focus on the estimation of the intensity s by model selection, starting with linear models in L2 (λ) that possess good √ approximating properties with respect to s.