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U-statistic with side information

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

Abstract

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

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    1. Yuichi Kitamura, 2006. "Empirical Likelihood Methods in Econometrics: Theory and Practice," Levine's Bibliography 321307000000000307, UCLA Department of Economics.
    2. 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.
    3. 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.
    4. Yuichi Kitamura, 2006. "Empirical Likelihood Methods in Econometrics: Theory and Practice," Cowles Foundation Discussion Papers 1569, Cowles Foundation for Research in Economics, Yale University.
    5. Yuichi Kitamura, 2006. "Empirical Likelihood Methods in Econometrics: Theory and Practice," CIRJE F-Series CIRJE-F-430, CIRJE, Faculty of Economics, University of Tokyo.
    6. 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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