U-statistic with side information

In this paper we study U-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 proposed U-statistics can have smaller asymptotic variance than the existing U-statistics in the literature. The proposed U-statistics can achieve asymptotic efficiency in a formal sense and their weak limits admit a convolution result. We also find that the corresponding U-likelihood ratio procedure, as well as the U-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 proposed U-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 the U-empirical likelihood ratio based method outperforms that of the normal approximation based method, in particular in the cases when the underlying distribution is skewed.