THE AS BEHAVIOR OF THE WEIGHTED EMPIRICAL PROCESS AND THE LIL FOR THE WEIGHTED TAIL EMPIRICAL PROCESS

• 成果类型: Article
• 署名作者: EINMAHL, JHJ
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176989800
• 发表日期: 1992
• 页码: 681-695
• 关键词: iterated logarithm; THEOREM

摘要

The tail empirical process is defined to be for each n is-an-element-of N, w(n)(t) = (n/k(n))1/2-alpha(n)(tk(n)/n), 0 less-than-or-equal-to t less-than-or-equal-to 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)}n=1infinity is a sequence of positive numbers with k(n)/n --> 0 and k(n) --> infinity. In this paper a complete description of the almost sure behavior of the weighted empirical process alpha(n)alpha(n)/q, where q is a weight function and {alpha(n)}n=1infinity 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)/loglog n --> infinity. 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.