The Annals of Probability

Asymptotic Normality of Statistics Based on the Convex Minorants of Empirical Distribution Functions

Piet Groeneboom and Ronald Pyke

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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 $\bar F_m = F_m \circ H^{-1}_N$.

Article information

Source
Ann. Probab., Volume 11, Number 2 (1983), 328-345.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993599

Digital Object Identifier
doi:10.1214/aop/1176993599

Mathematical Reviews number (MathSciNet)
MR690131

Zentralblatt MATH identifier
0521.62016

JSTOR
links.jstor.org

Subjects
Primary: 62E20: Asymptotic distribution theory
Secondary: 62G99: None of the above, but in this section 60J65: Brownian motion [See also 58J65]

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

Citation

Groeneboom, Piet; Pyke, Ronald. Asymptotic Normality of Statistics Based on the Convex Minorants of Empirical Distribution Functions. Ann. Probab. 11 (1983), no. 2, 328--345. doi:10.1214/aop/1176993599. https://projecteuclid.org/euclid.aop/1176993599


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