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The International Journal of Biostatistics

Ed. by Chambaz, Antoine / Hubbard, Alan E. / van der Laan, Mark J.

2 Issues per year


IMPACT FACTOR 2014: 0.741
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SCImago Journal Rank (SJR): 1.039
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Mathematical Citation Quotient 2013: 0.04

Targeted Maximum Likelihood Learning

Mark J. van der Laan1 / Daniel Rubin2

1Division of Biostatistics, School of Public Health, University of California, Berkeley

2University of California, Berkeley

Citation Information: The International Journal of Biostatistics. Volume 2, Issue 1, ISSN (Online) 1557-4679, DOI: 10.2202/1557-4679.1043, December 2006

Publication History

Published Online:
2006-12-28

Suppose one observes a sample of independent and identically distributed observations from a particular data generating distribution. Suppose that one is concerned with estimation of a particular pathwise differentiable Euclidean parameter. A substitution estimator evaluating the parameter of a given likelihood based density estimator is typically too biased and might not even converge at the parametric rate: that is, the density estimator was targeted to be a good estimator of the density and might therefore result in a poor estimator of a particular smooth functional of the density. In this article we propose a one step (and, by iteration, k-th step) targeted maximum likelihood density estimator which involves 1) creating a hardest parametric submodel with parameter epsilon through the given density estimator with score equal to the efficient influence curve of the pathwise differentiable parameter at the density estimator, 2) estimating epsilon with the maximum likelihood estimator, and 3) defining a new density estimator as the corresponding update of the original density estimator. We show that iteration of this algorithm results in a targeted maximum likelihood density estimator which solves the efficient influence curve estimating equation and thereby yields a locally efficient estimator of the parameter of interest, under regularity conditions. In particular, we show that, if the parameter is linear and the model is convex, then the targeted maximum likelihood estimator is often achieved in the first step, and it results in a locally efficient estimator at an arbitrary (e.g., heavily misspecified) starting density.We also show that the targeted maximum likelihood estimators are now in full agreement with the locally efficient estimating function methodology as presented in Robins and Rotnitzky (1992) and van der Laan and Robins (2003), creating, in particular, algebraic equivalence between the double robust locally efficient estimators using the targeted maximum likelihood estimators as an estimate of its nuisance parameters, and targeted maximum likelihood estimators. In addition, it is argued that the targeted MLE has various advantages relative to the current estimating function based approach. We proceed by providing data driven methodologies to select the initial density estimator for the targeted MLE, thereby providing data adaptive targeted maximum likelihood estimation methodology. We illustrate the method with various worked out examples.

Keywords: causal effect; cross-validation; efficient influence curve; estimating function; locally efficient estimation; loss function; maximum likelihood estimation; sieve; targeted maximum likelihood estimation; variable importance

Citing Articles

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[1]
M. J. Cohen
British Journal of Surgery, 2012, Volume 99, Number 4, Page 487
[2]
Thaddeus J. Haight, Yue Wang, Mark J. van der Laan, and Ira B. Tager
Computational Statistics & Data Analysis, 2010, Volume 54, Number 12, Page 3080
[4]
Samuel D. Lendle, Bruce Fireman, and Mark J. van der Laan
Journal of Clinical Epidemiology, 2013, Volume 66, Number 8, Page S91
[5]
MANABU KUROKI and ZHIHONG CAI
Scandinavian Journal of Statistics, 2011, Page no
[6]
Markus Frölich and Martin Huber
Journal of the American Statistical Association, 2014, Volume 109, Number 508, Page 1697
[7]
Mark J. van der Laan and Richard J. C. M. Starmans
Advances in Statistics, 2014, Volume 2014, Page 1
[8]
Zhiwei Zhang, Richard M. Kotz, Chenguang Wang, Shiling Ruan, and Martin Ho
Biometrics, 2013, Volume 69, Number 2, Page 318
[9]
Susan Gruber and Mark J. van der Laan
Biometrics, 2013, Volume 69, Number 1, Page 254
[10]
Alisa J. Stephens, Eric J. Tchetgen Tchetgen, and Victor De Gruttola
Statistics in Medicine, 2012, Volume 31, Number 10, Page 915
[11]
Zhiwei Zhang, Zhen Chen, James F. Troendle, and Jun Zhang
Biometrics, 2012, Volume 68, Number 3, Page 697
[12]
Iván Díaz Muñoz and Mark van der Laan
Biometrics, 2012, Volume 68, Number 2, Page 541
[13]
Hui Wang, Sherri Rose, and Mark J. van der Laan
Statistics & Probability Letters, 2011, Volume 81, Number 7, Page 792
[14]
Michael Rosenblum, Nicholas P. Jewell, Mark van der Laan, Stephen Shiboski, Ariane van der Straten, and Nancy Padian
Journal of the Royal Statistical Society: Series A (Statistics in Society), 2009, Volume 172, Number 2, Page 443
[15]
Oliver Bembom, Maya L. Petersen, Soo-Yon Rhee, W. Jeffrey Fessel, Sandra E. Sinisi, Robert W. Shafer, and Mark J. van der Laan
Statistics in Medicine, 2009, Volume 28, Number 1, Page 152
[16]
Oliver Bembom and Mark J. van der Laan
Statistics in Medicine, 2008, Volume 27, Number 19, Page 3689

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