Open Access
April 2016 Statistical inference for the mean outcome under a possibly non-unique optimal treatment strategy
Alexander R. Luedtke, Mark J. van der Laan
Ann. Statist. 44(2): 713-742 (April 2016). DOI: 10.1214/15-AOS1384


We consider challenges that arise in the estimation of the mean outcome under an optimal individualized treatment strategy defined as the treatment rule that maximizes the population mean outcome, where the candidate treatment rules are restricted to depend on baseline covariates. We prove a necessary and sufficient condition for the pathwise differentiability of the optimal value, a key condition needed to develop a regular and asymptotically linear (RAL) estimator of the optimal value. The stated condition is slightly more general than the previous condition implied in the literature. We then describe an approach to obtain root-$n$ rate confidence intervals for the optimal value even when the parameter is not pathwise differentiable. We provide conditions under which our estimator is RAL and asymptotically efficient when the mean outcome is pathwise differentiable. We also outline an extension of our approach to a multiple time point problem. All of our results are supported by simulations.


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Alexander R. Luedtke. Mark J. van der Laan. "Statistical inference for the mean outcome under a possibly non-unique optimal treatment strategy." Ann. Statist. 44 (2) 713 - 742, April 2016.


Received: 1 December 2014; Revised: 1 September 2015; Published: April 2016
First available in Project Euclid: 17 March 2016

zbMATH: 1338.62089
MathSciNet: MR3476615
Digital Object Identifier: 10.1214/15-AOS1384

Primary: 62G05
Secondary: 62N99

Keywords: Efficient estimator , non-regular inference , online estimation , optimal treatment , optimal value , Pathwise differentiability , semi parametric model

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.44 • No. 2 • April 2016
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