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May, 1983 Asymptotic Normality of Statistics Based on the Convex Minorants of Empirical Distribution Functions
Piet Groeneboom, Ronald Pyke
Ann. Probab. 11(2): 328-345 (May, 1983). DOI: 10.1214/aop/1176993599

Abstract

Let $F_n$ be the Uniform empirical distribution function. Write $\hat F_n$ for the (least) concave majorant of $F_n$, and let $\hat f_n$ denote the corresponding density. It is shown that $n \int^1_0 (\hat f_n(t) - 1)^2 dt$ is asymptotically standard normal when centered at $\log n$ and normalized by $(3 \log n)^{1/2}$. A similar result is obtained in the 2-sample case in which $\hat f_n$ is replaced by the slope of the convex minorant of $\bar F_m = F_m \circ H^{-1}_N$.

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Piet Groeneboom. Ronald Pyke. "Asymptotic Normality of Statistics Based on the Convex Minorants of Empirical Distribution Functions." Ann. Probab. 11 (2) 328 - 345, May, 1983. https://doi.org/10.1214/aop/1176993599

Information

Published: May, 1983
First available in Project Euclid: 19 April 2007

zbMATH: 0521.62016
MathSciNet: MR690131
Digital Object Identifier: 10.1214/aop/1176993599

Subjects:
Primary: 62E20
Secondary: 60J65 , 62G99

Keywords: Brownian bridge , concave majorant , convex minorant , Empirical distribution function , limit theorems , spacings , two-sample rank statistics

Rights: Copyright © 1983 Institute of Mathematical Statistics

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Vol.11 • No. 2 • May, 1983
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