## The Annals of Probability

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

#### 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

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