# 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.

1996). Weak Convergence and Empirical Processes. Springer, New York. -L. (1986). Asymptotically minimax estimators for distributions with increasing failure rate. Ann. Statist. 14 1113–1131. MR0856809  Wang, X. and Woodroofe, M. (2007). A Kiefer–Wolfowitz comparison theorem for Wicksell’s problem. Ann. Statist. 35. To appear.  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.