FUNCTIONAL LAWS OF THE ITERATED LOGARITHM FOR THE INCREMENTS OF EMPIRICAL AND QUANTILE PROCESSES

• 成果类型: Article
• 署名作者: DEHEUVELS, P; MASON, DM
• 署名单位: University of Delaware
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176989691
• 发表日期: 1992
• 页码: 1248-1287
• 关键词: strong limit-theorems; density estimators; large deviations

摘要

Let {alpha(n)(t), 0 less-than-or-equal-to t less-than-or-equal-to 1} and {beta(n)(t), 0 less-than-or-equal-to t less-than-or-equal-to 1} be the empirical and quantile processes generated by the first n observations from an i.i.d. sequence of uniformly distributed random variables on (0, 1). Let 0 < alpha(n) < 1 be a sequence of constants such that alpha(n) --> 0 as n --> infinity. We investigate the strong limiting behavior as n --> infinity of the increment functions {alpha(n)(t + alpha(n)s) - alpha(n)(t), 0 less-than-or-equal-to s less-than-or-equal-to 1} and {beta(n)(t + alpha(n)s) - beta(n)(t), 0 less-than-or-equal-to s less-than-or-equal-to 1}, where 0 less-than-or-equal-to t less-than-or-equal-to 1 - alpha(n). Under suitable regularity assumptions imposed upon alpha(n), we prove functional laws of the iterated logarithm for these increment functions and discuss statistical applications in the field of nonparametric estimation.