ON NORMAL APPROXIMATIONS TO U-STATISTICS

• 成果类型: Article
• 署名作者: Bentkus, Vidmantas; Jing, Bing-Yi; Zhou, Wang
• 署名单位: Vilnius University; Hong Kong University of Science & Technology; National University of Singapore
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/09-AOP474
• 发表日期: 2009
• 页码: 2174-2199
• 关键词: symmetric statistics; decompositions

摘要

Let X-1, ..., X-n be i.i.d. random observations. Let S = L + T be a U-statistic of order k >= 2 where L is a linear statistic having asymptotic normal distribution, and T is a stochastically smaller statistic. We show that the rate of convergence to normality for S can be simply expressed as the rate of convergence to normality for the linear part L plus a correction term, (varT) ln(2) (varT), under the condition ET2 < infinity. An optimal bound without this log factor is obtained under a lower moment assumption E vertical bar T vertical bar(alpha) < infinity for alpha < 2. Some other related results are also obtained in the paper. Our results extend, refine and yield a number of related-known results in the literature.