CV-TMLE for Nonpathwise Differentiable Target Parameters

  • Mark J. van der Laan
  • Aurélien Bibaut
  • Alexander R. Luedtke
Part of the Springer Series in Statistics book series (SSS)


TMLE has been developed for the construction of efficient substitution estimators of pathwise differentiable target parameters. Many parameters are nonpathwise differentiable such as a density or regression curves at a single point in a nonparametric model. In these cases one often uses a specific estimator under a specific smoothness assumptions for which it is possible to establish a limit distribution and thereby provide statistical inference. However, such estimators do not adapt to the true unknown smoothness of the data density and, as a consequence, can be easily outperformed by an adaptive estimator that is able to adapt to the underlying true smoothness.


  1. U. Einmahl, D.M. Mason, An empirical process approach to the uniform consistency of kernel-type function estimators. J. Theor. Probab. 13(1) 1–37 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  2. U. Einmahl, D.M. Mason, Uniform in bandwidth consistency of kernel-type function estimators. Ann. Stat. 33(3), 1380–1403 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. E.H. Kennedy, Z. Ma, M.D. McHugh, D.S. Small, Nonparametric methods for doubly robust estimation of continuous treatment effects. ArXiv e-prints (2015)Google Scholar
  4. E.A. Nadaraya, On estimating regression. Theory Probab. Appl. 9(1), 141–142 (1964)CrossRefzbMATHGoogle Scholar
  5. R. Neugebauer, M.J. van der Laan, Nonparametric causal effects based on marginal structural models. J. Stat. Plann. Infererence 137(2), 419–434 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. E. Parzen, On estimation of a probability density function and mode. Ann. Math. Stat. 33(3), 1065–1076 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  7. M. Rosenblatt, Remarks on some nonparametric estimates of a density function. Ann. Math. Stat. 27(3), 832–837 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  8. M. Rosenblum, M.J. van der Laan, Targeted maximum likelihood estimation of the parameter of a marginal structural model. Int. J. Biostat. 6(2), 19 (2010a)Google Scholar
  9. D.B. Rubin, M.J. van der Laan, Empirical efficiency maximization: improved locally efficient covariate adjustment in randomized experiments and survival analysis. Int. J. Biostat. 4(1), Article 5 (2008)Google Scholar
  10. A.W. van der Vaart, J.A. Wellner, A local maximal inequality under uniform entropy. Electron. J. Stat. 5, 192–203 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. G.S. Watson, Smooth regression analysis. Sankhyā Indian J. Stat. Ser. A 359–372 (1964)Google Scholar
  12. D. Wied, R. Weißbach, Consistency of the kernel density estimator: a survey. Stat. Pap. 53(1), 1–21 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Mark J. van der Laan
    • 1
  • Aurélien Bibaut
    • 2
  • Alexander R. Luedtke
    • 3
  1. 1.Division of Biostatistics and Department of StatisticsUniversity of CaliforniaBerkeleyUSA
  2. 2.Division of BiostatisticsUniversity of CaliforniaBerkeleyUSA
  3. 3.Vaccine and Infectious Disease DivisionFred Hutchinson Cancer Research CenterSeattleUSA

Personalised recommendations