## The Annals of Probability

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

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

Piet Groeneboom and Ronald Pyke

#### 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