Abstract
The objective of this chapter is to describe how the TMLE framework can be generalized to explicitly utilize higher-order rather than first-order asymptotic representations. The practical significance of this is to provide guidelines for constructing estimators that have sound behavior in finite samples and are asymptotically efficient under less restrictive conditions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
P.J. Bickel, On adaptive estimation. Ann. Stat. 10, 647–671 (1982)
P.J. Bickel, C.A.J. Klaassen, Y. Ritov, J. Wellner, Efficient and Adaptive Estimation for Semiparametric Models (Springer, Berlin, Heidelberg, New York, 1997b)
M. Carone, I. Díaz, M.J. van der Laan, Higher-order targeted minimum loss-based estimation. Technical Report, Division of Biostatistics, University of California, Berkeley
I. Díaz, M. Carone, M.J. van der Laan, Second-order inference for the mean of a variable missing at random. Int. J. Biostat. 12(1), 333–349 (2016)
R.D. Gill, M.J. van der Laan, J.A. Wellner, Inefficient estimators of the bivariate survival function for three models. Ann. l’Institut Henri Poincaré 31(3), 545–597 (1995)
S. Gruber, M.J. van der Laan, An application of collaborative targeted maximum likelihood estimation in causal inference and genomics. Int. J. Biostat. 6(1) (2010a)
S. Gruber, M.J. van der Laan, Targeted minimum loss based estimator that outperforms a given estimator. Int. J. Biostat. 8(1), (2012b)
I.A. Ibragimov, R.Z. Khasminskii, Statistical Estimation (Springer, Berlin, 1981)
S.D. Lendle, B. Fireman, M.J. van der Laan, Balancing score adjusted targeted minimum loss-based estimation. Technical Report, Division of Biostatistics, University of California, Berkeley (2013)
B.Y. Levit, On the efficiency of a class of non-parametric estimates. Theory Probab. Appl. 20(4), 723–740 (1975)
L. Li, E. Tchetgen Tchetgen, A.W. van der Vaart, J.M. Robins, Higher order inference on a treatment effect under low regularity conditions. Stat. Probab. Lett. 81(7), 821–828 (2011)
J. Pfanzagl, Contributions to a General Asymptotic Statistical Theory (Springer, Berlin, 1982)
J. Pfanzagl, Asymptotic Expansions for General Statistical Models, vol. 31 (Springer, Berlin, 1985)
E.C Polley, S. Rose, M.J. van der Laan, Super-learning, in Targeted Learning: Causal Inference for Observational and Experimental Data, ed. by M.J. van der Laan, S. Rose (Springer, Berlin, Heidelberg, New York, 2011)
J.M. Robins, L. Li, E. Tchetgen Tchetgen, A.W. van der Vaart, Higher order influence functions and minimax estimation of nonlinear functionals, in Probability and Statistics: Essays in Honor of David A. Freedman, (Institute of Mathematical Statistics, 2008a), pp. 335–421
J.M. Robins, L. Li, E. Tchetgen Tchetgen, A.W. van der Vaart, Quadratic Semiparametric Von Mises calculus. Metrika 69(2–3), 227–247 (2009)
D.B. Rubin, M.J. van der Laan, Targeted ANCOVA estimator in RCTs, in Targeted Learning (Springer, Berlin, 2011), pp. 201–215
O.M. Stitelman, M.J. van der Laan, Collaborative targeted maximum likelihood for time-to-event data. Int. J. Biostat. 6(1), Article 21 (2010)
M.J. van der Laan, Targeted estimation of nuisance parameters to obtain valid statistical inference. Int. J. Biostat. 10(1), 29–57 (2014b)
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)
M.J. van der Laan, S. Gruber, Collaborative double robust penalized targeted maximum likelihood estimation. Int. J. Biostat. 6(1), Article 17 (2010)
M.J. van der Laan, E.C. Polley, A.E. Hubbard, Super learner. Stat. Appl. Genet. Mol. 6(1), Article 25 (2007)
M.J. van der Laan, S. Rose, Targeted Learning: Causal Inference for Observational and Experimental Data (Springer, Berlin, Heidelberg, New York, 2011)
M.J. van der Laan, D.B. Rubin, Targeted maximum likelihood learning. Int. J. Biostat. 2(1), Article 11 (2006)
A.W. van der Vaart, Higher order tangent spaces and influence functions. Stat. Sci. 29(4), 679–686 (2014)
A.W. van der Vaart, J.A. Wellner, Weak Convergence and Empirical Processes (Springer, Berlin, Heidelberg, New York, 1996)
A.W. van der Vaart, S. Dudoit, M.J. van der Laan, Oracle inequalities for multi-fold cross-validation. Stat. Decis. 24(3), 351–371 (2006)
H. Wang, S. Rose, M.J. van der Laan, Finding quantitative trait loci genes with collaborative targeted maximum likelihood learning. Stat. Probab. Lett. 81(7), 792–796 (2011a)
W. Zheng, M.J. van der Laan, Asymptotic theory for cross-validated targeted maximum likelihood estimation. Technical Report, Division of Biostatistics, University of California, Berkeley (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
Carone, M., Díaz, I., van der Laan, M.J. (2018). Higher-Order Targeted Loss-Based Estimation. In: Targeted Learning in Data Science. Springer Series in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-65304-4_26
Download citation
DOI: https://doi.org/10.1007/978-3-319-65304-4_26
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-65303-7
Online ISBN: 978-3-319-65304-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)