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U-statistic with side information


  • Yuan, Ao
  • He, Wenqing
  • Wang, Binhuan
  • Qin, Gengsheng


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.

Suggested Citation

  • Yuan, Ao & He, Wenqing & Wang, Binhuan & Qin, Gengsheng, 2012. "U-statistic with side information," Journal of Multivariate Analysis, Elsevier, vol. 111(C), pages 20-38.
  • Handle: RePEc:eee:jmvana:v:111:y:2012:i:c:p:20-38
    DOI: 10.1016/j.jmva.2012.04.008

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    References listed on IDEAS

    1. Jing Qin & Biao Zhang, 2005. "Marginal likelihood, conditional likelihood and empirical likelihood: Connections and applications," Biometrika, Biometrika Trust, vol. 92(2), pages 251-270, June.
    2. Gengsheng Qin & Xiao-Hua Zhou, 2006. "Empirical Likelihood Inference for the Area under the ROC Curve," Biometrics, The International Biometric Society, vol. 62(2), pages 613-622, June.
    3. Gianfranco Adimari, 1997. "Empirical Likelihood Type Confidence Intervals Under Random Censorship," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 49(3), pages 447-466, September.
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    Cited by:

    1. Zhang, Qi & Mclellan, Benjamin C. & Tezuka, Tetsuo & Ishihara, Keiichi N., 2013. "A methodology for economic and environmental analysis of electric vehicles with different operational conditions," Energy, Elsevier, vol. 61(C), pages 118-127.
    2. Abrell, Jan & Rausch, Sebastian, 2016. "Cross-country electricity trade, renewable energy and European transmission infrastructure policy," Journal of Environmental Economics and Management, Elsevier, vol. 79(C), pages 87-113.
    3. Zhang, Qi & Mclellan, Benjamin C. & Tezuka, Tetsuo & Ishihara, Keiichi N., 2013. "An integrated model for long-term power generation planning toward future smart electricity systems," Applied Energy, Elsevier, vol. 112(C), pages 1424-1437.
    4. Yuan, Ao & Zheng, Yanxun & Huang, Peng & Tan, Ming T., 2016. "A nonparametric test for the evaluation of group sequential clinical trials with covariate information," Journal of Multivariate Analysis, Elsevier, vol. 152(C), pages 82-99.


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