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.Comment: Published at http://dx.doi.org/10.1214/15-AOS1384 in the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
