The Annals of Probability

The A.S. Behavior of the Weighted Empirical Process and the LIL for the Weighted Tail Empirical Process

John H. J. Einmahl

Abstract

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.

Article information

Source
Ann. Probab., Volume 20, Number 2 (1992), 681-695.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176989800

Digital Object Identifier
doi:10.1214/aop/1176989800

Mathematical Reviews number (MathSciNet)
MR1159568

Zentralblatt MATH identifier
0754.60028

JSTOR