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Asymptotic normality of quadratic estimators

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  • Robins, James M.
  • Li, Lingling
  • Tchetgen, Eric Tchetgen
  • van der Vaart, Aad

Abstract

We prove conditional asymptotic normality of a class of quadratic U-statistics that are dominated by their degenerate second order part and have kernels that change with the number of observations. These statistics arise in the construction of estimators in high-dimensional semi- and non-parametric models, and in the construction of nonparametric confidence sets. This is illustrated by estimation of the integral of a square of a density or regression function, and estimation of the mean response with missing data. We show that estimators are asymptotically normal even in the case that the rate is slower than the square root of the observations.

Suggested Citation

  • Robins, James M. & Li, Lingling & Tchetgen, Eric Tchetgen & van der Vaart, Aad, 2016. "Asymptotic normality of quadratic estimators," Stochastic Processes and their Applications, Elsevier, vol. 126(12), pages 3733-3759.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:12:p:3733-3759
    DOI: 10.1016/j.spa.2016.04.005
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    References listed on IDEAS

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    1. Bhattacharya, Rabi N. & Ghosh, Jayanta K., 1992. "A class of U-statistics and asymptotic normality of the number of k-clusters," Journal of Multivariate Analysis, Elsevier, vol. 43(2), pages 300-330, November.
    2. de Jong, Peter, 1990. "A central limit theorem for generalized multilinear forms," Journal of Multivariate Analysis, Elsevier, vol. 34(2), pages 275-289, August.
    3. Whitney K. Newey & Fushing Hsieh & James M. Robins, 2004. "Twicing Kernels and a Small Bias Property of Semiparametric Estimators," Econometrica, Econometric Society, vol. 72(3), pages 947-962, May.
    4. Mikosch, T., 1993. "A Weak Invariance Principle for Weighted U-Statistics with Varying Kernels," Journal of Multivariate Analysis, Elsevier, vol. 47(1), pages 82-102, October.
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    Cited by:

    1. Vladimir Koltchinskii & Mayya Zhilova, 2021. "Estimation of Smooth Functionals of Location Parameter in Gaussian and Poincaré Random Shift Models," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(2), pages 569-596, August.

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