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

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

**Read or Download Asymptotics: particles, processes, and inverse problems: festschrift for Piet Groeneboom PDF**

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**Example text**

Jongbloed, G. and Wellner, J. A. (2001). Estimation of a convex function: characterizations and asymptotic theory. Ann. Statist. 29 1653–1698. MR1891742 [12] Hall, C. A. (1968). On error bounds for spline interpolation. J. Approximation Theory 1 209–218. MR0239324 [13] Hall, C. A. and Meyer, W. W. (1976). Optimal error bounds for cubic spline interpolation. J. Approximation Theory 16 105–122. MR0397247 [14] Kiefer, J. and Wolfowitz, J. (1976). Asymptotically minimax estimation of concave and convex distribution functions.

Asymptotically minimax estimation of concave and convex distribution functions. Z. Wahrsch. Verw. Gebiete 34 73–85. MR0397974 [15] Kiefer, J. and Wolfowitz, J. (1977). Asymptotically minimax estimation of concave and convex distribution functions. II. 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.

This implies the claimed bound. Main steps: A. By Marshall’s lemma, for any concave function h, Fn − h ≤ Fn − h . (k ) B. PF (An ) ≡ PF {Ln n is concave on [0, ∞)} ր 1 as n → ∞ if kn ≡ (C0 β1 (F )× 1/3 n/ log n) for some absolute constant C0 . C. On the event An , it follows from Marshall’s lemma (step A) that n) n) + L(k − Fn Fn − Fn = Fn − L(k n n (kn ) (kn ) ≤ Fn − Ln + Ln − Fn n) = 2 Fn − L(k n n) = 2 Fn − L(k − (F − L(kn ) ) + F − L(kn ) ) n n) ≤ 2 Fn − L(k − (F − L(kn ) ) + 2 F − L(kn ) n ≡ 2(Dn + En ).