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

ISBN-10: 0940600714

ISBN-13: 9780940600713

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

**Example text**

U Applying this second formula to g = Yn − Y yields the relation aj (1) EYn −Y (t) = aj−1 Eg(1) (t)d(Fn − F )(u). u t (1) (1) Now gu is absolutely continuous with gu (t) = 0 gu (s)ds where gu (t) = 1[t≥u] , so by de Boor [5], (17) on page 56 (recalling that our C = I4 of de Boor), = gu(1) − (C[gu ])(1) Eg(1) u ≤ (19/4)dist(gu(1) , $3 ) ≤ (19/4)dist(gu(1) , $2 ) ≤ (19/4)ω(gu(1), |a|) ≤ (19/4) ≤ 5. 1 in section 3. 1 we find that (1) P r |EYn −Y (al )| > δn p3n nδn2 p6n /2 + pn (5/3)δn p3n /f (a∗j ) ≤ 2 exp − 50p3n /f (a∗j )2 = 2 exp − nδn2 f 2 (a∗j )p3n 100 + (10/3)pnf (a∗j )δn = 2 exp − (100)−1 nδn2 f 2 (a∗j )p3n 1 + (1/30)pnδn f (a∗j ) .

Moreover sδ − sδ 2 2 = θ∆(δ, δ ). 8). This result implies that, if we want to use the squared L2 -norm as a loss function, whatever the choice of our estimator there is no hope to ﬁnd risk bounds that are independent of the L∞ -norm of the underlying intensity, even if this intensity belongs to a ﬁnite-dimensional aﬃne space. This provides an additional motivation for the introduction of loss functions based on the distance H. 3. 1. Some notations Throughout this paper, we observe a Poisson process X on X with unknown mean measure µ belonging to the metric space (Q+ (X ), H) and have at hand some ref⊥ erence measure λ on X so that µ = µs + µ⊥ with µs ∈ Qλ , s ∈ L+ 1 (λ) and µ Model selection for Poisson processes 41 orthogonal to λ.

Given two distinct points t, u ∈ S there exists a test ψt,u between t and u which satisﬁes sup {µ∈Q+ (X ) | H(µ,µt )≤H(t,u)/4} Pµ [ψt,u (X) = u] ≤ exp − H 2 (t, u) − η 2 (t) + η 2 (u) /4 , L. 5) Pµ [ψt,u (X) = u] ≤ exp 16H 2 (µ, µt ) + η 2 (t) − η 2 (u) /4 . To build a T-estimator, we proceed as follows. We consider a family of tests ψt,u indexed by the two-points subsets {t, u} of S with t = u that satisfy the conclusions of Proposition 1 and we set Rt = {u ∈ S, u = t | ψt,u (X) = u} for each t ∈ S.

### 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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