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Characterization of parameters with a mixed bias property

Author

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  • A Rotnitzky
  • E Smucler
  • J M Robins

Abstract

SummaryWe study a class of parameters with the so-called mixed bias property. For parameters with this property, the bias of the semiparametric efficient one-step estimator is equal to the mean of the product of the estimation errors of two nuisance functions. In nonparametric models, parameters with the mixed bias property admit so-called rate doubly robust estimators, i.e., estimators that are consistent and asymptotically normal when one succeeds in estimating both nuisance functions at sufficiently fast rates, with the possibility of trading off slower rates of convergence for the estimator of one of the nuisance functions against faster rates for the estimator of the other nuisance function. We show that the class of parameters with the mixed bias property strictly includes two recently studied classes of parameters which, in turn, include many parameters of interest in causal inference. We characterize the form of parameters with the mixed bias property and of their influence functions. Furthermore, we derive two functional loss functions, each being minimized at one of the two nuisance functions. These loss functions can be used to derive loss-based penalized estimators of the nuisance functions.

Suggested Citation

  • A Rotnitzky & E Smucler & J M Robins, 2021. "Characterization of parameters with a mixed bias property," Biometrika, Biometrika Trust, vol. 108(1), pages 231-238.
    • Handle: RePEc:oup:biomet:v:108:y:2021:i:1:p:231-238.
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    File URL: http://hdl.handle.net/10.1093/biomet/asaa054
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    Citations

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    Cited by:

    1. Jikai Jin & Vasilis Syrgkanis, 2024. "Structure-agnostic Optimality of Doubly Robust Learning for Treatment Effect Estimation," Papers 2402.14264, arXiv.org, revised Mar 2024.
    2. Xingyu Chen & Lin Liu & Rajarshi Mukherjee, 2024. "Method-of-Moments Inference for GLMs and Doubly Robust Functionals under Proportional Asymptotics," Papers 2408.06103, arXiv.org.
    3. Y Cui & E J Tchetgen Tchetgen, 2024. "Selective machine learning of doubly robust functionals," Biometrika, Biometrika Trust, vol. 111(2), pages 517-535.
    4. Liu, Lin & Mukherjee, Rajarshi & Robins, James M., 2024. "Assumption-lean falsification tests of rate double-robustness of double-machine-learning estimators," Journal of Econometrics, Elsevier, vol. 240(2).
    5. Lin Liu & Chang Li, 2023. "New $\sqrt{n}$-consistent, numerically stable higher-order influence function estimators," Papers 2302.08097, arXiv.org.
    6. Christoph Breunig & Ruixuan Liu & Zhengfei Yu, 2022. "Double Robust Bayesian Inference on Average Treatment Effects," Papers 2211.16298, arXiv.org, revised Oct 2024.
    7. Isaac Meza & Rahul Singh, 2021. "Nested Nonparametric Instrumental Variable Regression: Long Term, Mediated, and Time Varying Treatment Effects," Papers 2112.14249, arXiv.org, revised Mar 2024.

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