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U-statistics with conditional kernels for incomplete data models

Author

Listed:
  • Ao Yuan

    (Georgetown University)

  • Mihai Giurcanu

    (University of Florida)

  • George Luta

    (Georgetown University)

  • Ming T. Tan

    (Georgetown University)

Abstract

For incomplete data models, the classical U-statistic estimator of a functional parameter of the underlying distribution cannot be computed directly since the data are not fully observed. To estimate such a functional parameter, we propose a U-statistic using a substitution estimator of the conditional kernel given the observed data. This kernel estimator is obtained by substituting the non-parametric maximum likelihood estimator for the underlying distribution function in the expression of the conditional kernel. We study the asymptotic properties of the proposed U-statistic for several incomplete data models, and in a simulation study, we assess the finite sample performance of the Mann–Whitney U-statistic with conditional kernel in the current status model. The analysis of a real-world data set illustrates the application of the proposed methods in practice.

Suggested Citation

  • Ao Yuan & Mihai Giurcanu & George Luta & Ming T. Tan, 2017. "U-statistics with conditional kernels for incomplete data models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(2), pages 271-302, April.
    • Handle: RePEc:spr:aistmt:v:69:y:2017:i:2:d:10.1007_s10463-015-0537-6
    DOI: 10.1007/s10463-015-0537-6
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    References listed on IDEAS

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    1. Somnath Datta & Dipankar Bandyopadhyay & Glen A. Satten, 2010. "Inverse Probability of Censoring Weighted U‐statistics for Right‐Censored Data with an Application to Testing Hypotheses," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(4), pages 680-700, December.
    2. Bose, Arup & Sen, Arusharka, 2002. "Asymptotic Distribution of the Kaplan-Meier U-Statistics," Journal of Multivariate Analysis, Elsevier, vol. 83(1), pages 84-123, October.
    3. Patrick J. Heagerty & Yingye Zheng, 2005. "Survival Model Predictive Accuracy and ROC Curves," Biometrics, The International Biometric Society, vol. 61(1), pages 92-105, March.
    4. Tressou, Jessica, 2006. "Nonparametric Modeling of the Left Censorship of Analytical Data in Food Risk Assessment," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1377-1386, December.
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    Cited by:

    1. Mihai C. Giurcanu & Theodore G. Karrison, 2022. "Nonparametric inference in the accelerated failure time model using restricted means," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 28(1), pages 23-39, January.
    2. Salim Bouzebda & Amel Nezzal & Tarek Zari, 2022. "Uniform Consistency for Functional Conditional U -Statistics Using Delta-Sequences," Mathematics, MDPI, vol. 11(1), pages 1-39, December.
    3. Salim Bouzebda & Thouria El-hadjali & Anouar Abdeldjaoued Ferfache, 2023. "Uniform in Bandwidth Consistency of Conditional U-statistics Adaptive to Intrinsic Dimension in Presence of Censored Data," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(2), pages 1548-1606, August.

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