Let $\{\alpha_n(t), 0 \leq t \leq 1\}$ and $\{\beta_n(t), 0 \leq t \leq 1\}$ be the empirical and quantile processes generated by the first $n$ observations from an i.i.d. sequence of uniformly distributed random variables on (0,1). Let $0 < a_n < 1$ be a sequence of constants such that $a_n \rightarrow 0$ as $n \rightarrow \infty$. We investigate the strong limiting behavior as $n \rightarrow \infty$ of the increment functions $\{\alpha_n(t + a_ns) - \alpha_n(t), 0 \leq s \leq 1\}$ and $\{\beta_n(t + a_ns) - \beta_n(t), 0 \leq s \leq 1\},$ where $0 \leq t \leq 1 - a_n$. Under suitable regularity assumptions imposed upon $a_n$, we prove functional laws of the iterated logarithm for these increment functions and discuss statistical applications in the field of nonparametric estimation.
Publié le : 1992-07-14
Classification:
Functional limit laws,
laws of the iterated logarithm,
empirical processes,
quantile processes,
order statistics,
nonparametric estimation,
density estimation,
nearest neighbor estimates,
60F15,
60F17,
62G05
@article{1176989691,
author = {Deheuvels, Paul and Mason, David M.},
title = {Functional Laws of the Iterated Logarithm for the Increments of Empirical and Quantile Processes},
journal = {Ann. Probab.},
volume = {20},
number = {4},
year = {1992},
pages = { 1248-1287},
language = {en},
url = {http://dml.mathdoc.fr/item/1176989691}
}
Deheuvels, Paul; Mason, David M. Functional Laws of the Iterated Logarithm for the Increments of Empirical and Quantile Processes. Ann. Probab., Tome 20 (1992) no. 4, pp. 1248-1287. http://gdmltest.u-ga.fr/item/1176989691/