HAL Estimator of the Efficient Influence Curve

Chapter
Part of the Springer Series in Statistics book series (SSS)

Abstract

The construction of an efficient estimator of a pathwise differentiable target parameter \(\varPsi: \mathcal{M}\rightarrow \mathbb{R}\) relies on the ability to evaluate its canonical gradient D(P) at an initial estimator P of P0 based on an original i.i.d. sample from P0. The efficient influence curve D(P) is defined as the canonical gradient of the pathwise derivative of the target parameter along parametric submodels through P. It is always possible to represent the pathwise derivative of the target parameter along a parametric submodel as a covariance of a gradient D(P) ∈ L02(P) with the score of the submodel. The canonical gradient is now defined as the projection of this gradient on the tangent space at P, where the tangent space is defined as the closure of the linear span of all scores one can generate with a parametric submodel through P.

References

  1. C.E. Frangakis, T. Qian, Z. Wu, I. Diaz, Deductive derivation and Turing-computerization of semiparametric efficient estimation. Biometrics 71(4), 867–874 (2015)MathSciNetCrossRefMATHGoogle Scholar
  2. A..R Luedtke, M. Carone, M.J. van der Laan, Discussion of deductive derivation and turing-computerization of semiparametric efficient estimation by Frangakis et al. Biometrics 71(4), 875–879 (2015a)Google Scholar
  3. M.J. van der Laan, A generally efficient targeted minimum loss based estimator. Int. J. Biostat. 13(2), 1106–1118 (2017)Google Scholar
  4. M.J. van der Laan, S. Dudoit, Unified cross-validation methodology for selection among estimators and a general cross-validated adaptive epsilon-net estimator: finite sample oracle inequalities and examples. Technical Report, Division of Biostatistics, University of California, Berkeley (2003)Google Scholar
  5. M.J. van der Laan, M. Carone, A.R. Luedtke, Computerizing efficient estimation of a pathwise differentiable target parameter. Technical Report, Division of Biostatistics, University of California, Berkeley (2015)Google Scholar
  6. A.W. van der Vaart, J.A. Wellner, A local maximal inequality under uniform entropy. Electron. J. Stat. 5, 192–203 (2011)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Division of Biostatistics and Department of StatisticsUniversity of California, BerkeleyBerkeleyUSA

Personalised recommendations