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One-Step Targeted Minimum Loss-based Estimation Based on Universal Least Favorable One-Dimensional Submodels

Author

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  • van der Laan Mark

    (Division of Biostatistics, University of California, Berkeley, CA, USA)

  • Gruber Susan

    (Department of Population Medicine, Harvard Medical School and Harvard Pilgrim Healthcare Institute, Boston, MA, USA)

Abstract

Consider a study in which one observes n independent and identically distributed random variables whose probability distribution is known to be an element of a particular statistical model, and one is concerned with estimation of a particular real valued pathwise differentiable target parameter of this data probability distribution. The targeted maximum likelihood estimator (TMLE) is an asymptotically efficient substitution estimator obtained by constructing a so called least favorable parametric submodel through an initial estimator with score, at zero fluctuation of the initial estimator, that spans the efficient influence curve, and iteratively maximizing the corresponding parametric likelihood till no more updates occur, at which point the updated initial estimator solves the so called efficient influence curve equation. In this article we construct a one-dimensional universal least favorable submodel for which the TMLE only takes one step, and thereby requires minimal extra data fitting to achieve its goal of solving the efficient influence curve equation. We generalize these to universal least favorable submodels through the relevant part of the data distribution as required for targeted minimum loss-based estimation. Finally, remarkably, given a multidimensional target parameter, we develop a universal canonical one-dimensional submodel such that the one-step TMLE, only maximizing the log-likelihood over a univariate parameter, solves the multivariate efficient influence curve equation. This allows us to construct a one-step TMLE based on a one-dimensional parametric submodel through the initial estimator, that solves any multivariate desired set of estimating equations.

Suggested Citation

  • van der Laan Mark & Gruber Susan, 2016. "One-Step Targeted Minimum Loss-based Estimation Based on Universal Least Favorable One-Dimensional Submodels," The International Journal of Biostatistics, De Gruyter, vol. 12(1), pages 351-378, May.
    • Handle: RePEc:bpj:ijbist:v:12:y:2016:i:1:p:351-378:n:19
    DOI: 10.1515/ijb-2015-0054
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    References listed on IDEAS

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    1. Stitelman Ori M & van der Laan Mark J., 2010. "Collaborative Targeted Maximum Likelihood for Time to Event Data," The International Journal of Biostatistics, De Gruyter, vol. 6(1), pages 1-46, June.
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    Cited by:

    1. Frölich, Markus & Huber, Martin & Wiesenfarth, Manuel, 2017. "The finite sample performance of semi- and non-parametric estimators for treatment effects and policy evaluation," Computational Statistics & Data Analysis, Elsevier, vol. 115(C), pages 91-102.
    2. Iván Díaz & Elizabeth Colantuoni & Daniel F. Hanley & Michael Rosenblum, 2019. "Improved precision in the analysis of randomized trials with survival outcomes, without assuming proportional hazards," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 25(3), pages 439-468, July.

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