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

J.H.J. Einmahl

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    Abstract

    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.
    Original languageEnglish
    Pages (from-to)681-695
    Number of pages15
    JournalThe Annals of Probability
    Volume20
    Issue number2
    DOIs
    Publication statusPublished - 1992

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