title: U-statistic with side information.
abstract: In this paper we studyU-statistics with side information incorporated using the method of empirical likelihood. Some basic properties of the proposed statistics are investigated. We find that by implementing the side information properly, the proposedU-statistics can have smaller asymptotic variance than the existingU-statistics in the literature. The proposedU-statistics can achieve asymptotic efficiency in a formal sense and their weak limits admit a convolution result. We also find that the correspondingU-likelihood ratio procedure, as well as theU-empirical likelihood based confidence interval construction, do not benefit from incorporating side information, a result that is consistent with the result under the standard empirical likelihood ratio procedure. The impact of incorrect side information implementation in the proposedU-statistics is also explored. Simulation studies are conducted to assess the finite sample performance of the proposed method. The numerical results show that with side information implemented, the deduction of asymptotic variance can be substantial in some cases, and the coverage probability of the confidence interval using theU-empirical likelihood ratio based method outperforms that of the normal approximation based method, in particular in the cases when the underlying distribution is skewed.
authors.affiliation: Howard University, Washington DC 20059, USA.
authors.forename: Ao
authors.initials: A
authors.lastname: Yuan
authors.forename: Wenqing
authors.initials: W
authors.lastname: He
authors.forename: Binhuan
authors.initials: B
authors.lastname: Wang
authors.forename: Gengsheng
authors.initials: G
authors.lastname: Qin
keywords: Efficiency; Information bound; Side information; U-statistic