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
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Extra resources for Asymptotics: particles, processes and inverse problems. Festschrift for Piet Groeneboom
Example text
7) exp −2H 2 (sδ , sδ ) ≥ exp[−2/7]. (δ,δ )∈C Proof. The first equality is clear. Let us then observe that our assumptions on g imply that 1 − g 2 (x)/7 ≤ 1 − g 2 (x)/4 ≤ 1 − g 2 (x)/8, hence, since the functions gj have disjoint supports and are translates of g, D −1 D −1 2 |δj − δj | H (sδ , sδ ) = (2a) 1 + g(x)/2 − 2 dx 0 j=1 D −1 D = a−1 1 − g(x)/2 |δj − δj | j=1 1− 1 − g 2 (x)/4 dx = c∆(δ, δ ), 0 with 1/8 ≤ c ≤ 1/7. The conclusions follow. Corollary 1. 8) sup Es s∈SD sˆ − s 2 2 ≥ (DL/24) exp[−2/7].
The conclusions follow. Corollary 1. 8) sup Es s∈SD sˆ − s 2 2 ≥ (DL/24) exp[−2/7]. Proof. Let us set θ = 2L/3 ≥ D and apply the construction of Lemma 2 with g(x) = D/θ 1l[0,1/D) , hence a = θ−1 . This results in the set SD with sδ ∞ ≤ θ 1 + (1/2) D/θ ≤ 3θ/2 = L for all δ ∈ D as required. 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.
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