TY - JOUR

T1 - The almost sure behavior of the weighted empirical process and the law of the iterated logarithm for the weighted tail empirical process

AU - Einmahl, J.H.J.

PY - 1992

Y1 - 1992

N2 - The tail empirical process is defined to be for each n ¿ N, wn(t) = (n/kn)1/2an(tkn/n), 0 = t = 1, where an is the empirical process based on the first n of a sequence of independent uniform (0,1) random variables and {kn}8 n=1 is a sequence of positive numbers with kn/n ¿ 0 and kn ¿ 8. In this paper a complete description of the almost sure behavior of the weighted empirical process anan/q, where q is a weight function and {an}8 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 wn/q, provided kn/loglog n ¿ 8. These results unify and generalize several results in the literature. Moreover, a characterization of the central limit theorem behavior of wn/q is presented. That result is applied to the construction of asymptotic confidence bands for intermediate quantiles from an arbitrary continuous distribution.

AB - The tail empirical process is defined to be for each n ¿ N, wn(t) = (n/kn)1/2an(tkn/n), 0 = t = 1, where an is the empirical process based on the first n of a sequence of independent uniform (0,1) random variables and {kn}8 n=1 is a sequence of positive numbers with kn/n ¿ 0 and kn ¿ 8. In this paper a complete description of the almost sure behavior of the weighted empirical process anan/q, where q is a weight function and {an}8 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 wn/q, provided kn/loglog n ¿ 8. These results unify and generalize several results in the literature. Moreover, a characterization of the central limit theorem behavior of wn/q is presented. That result is applied to the construction of asymptotic confidence bands for intermediate quantiles from an arbitrary continuous distribution.

U2 - 10.1214/aop/1176989800

DO - 10.1214/aop/1176989800

M3 - Article

VL - 20

SP - 681

EP - 695

JO - The Annals of Probability

JF - The Annals of Probability

SN - 0091-1798

IS - 2

ER -