The A.S. Behavior of the Weighted Empirical Process and the LIL for the Weighted Tail Empirical Process
Einmahl, John H. J.
Ann. Probab., Tome 20 (1992) no. 4, p. 681-695 / Harvested from Project Euclid
The tail empirical process is defined to be for each $n \in \mathbb{N}, w_n(t) = (n/k_n)^{1/2}\alpha_n(tk_n/n), 0 \leq t \leq 1$, where $\alpha_n$ is the empirical process based on the first $n$ of a sequence of independent uniform (0,1) random variables and $\{k_n\}^\infty_{n=1}$ is a sequence of positive numbers with $k_n/n \rightarrow 0$ and $k_n \rightarrow \infty$. In this paper a complete description of the almost sure behavior of the weighted empirical process $a_n\alpha_n/q$, where $q$ is a weight function and $\{a_n\}^\infty_{n=1}$ is a sequence of positive numbers, is established as well as a characterization of the law of the iterated logarithm behavior of the weighted tail empirical process $w_n/q$, provided $k_n/\log\log n \rightarrow \infty$. These results unify and generalize several results in the literature. Moreover, a characterization of the central limit theorem behavior of $w_n/q$ is presented. That result is applied to the construction of asymptotic confidence bands for intermediate quantiles from an arbitrary continuous distribution.
Publié le : 1992-04-14
Classification:  Confidence band,  empirical process,  intermediate quantiles,  strong and weak limit theorems,  tail empirical process,  weight-function,  60F15,  60F05,  62G15,  62G30
@article{1176989800,
     author = {Einmahl, John H. J.},
     title = {The A.S. Behavior of the Weighted Empirical Process and the LIL for the Weighted Tail Empirical Process},
     journal = {Ann. Probab.},
     volume = {20},
     number = {4},
     year = {1992},
     pages = { 681-695},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1176989800}
}
Einmahl, John H. J. The A.S. Behavior of the Weighted Empirical Process and the LIL for the Weighted Tail Empirical Process. Ann. Probab., Tome 20 (1992) no. 4, pp.  681-695. http://gdmltest.u-ga.fr/item/1176989800/