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{1176993599,
author = {Groeneboom, Piet and Pyke, Ronald},
title = {Asymptotic Normality of Statistics Based on the Convex Minorants of Empirical Distribution Functions},
journal = {Ann. Probab.},
volume = {11},
number = {4},
year = {1983},
pages = { 328-345},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993599}
}
Groeneboom, Piet; Pyke, Ronald. Asymptotic Normality of Statistics Based on the Convex Minorants of Empirical Distribution Functions. Ann. Probab., Tome 11 (1983) no. 4, pp. 328-345. http://gdmltest.u-ga.fr/item/1176993599/