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

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.