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