  Coarse structural nested mean models are used to estimate treatment effects
from longitudinal observational data. Coarse structural nested mean models lead
to a large class of estimators. It turns out that estimates and standard errors
may differ considerably within this class. We prove that, under additional
assumptions, there exists an explicit solution for the optimal estimator within
the class of coarse structural nested mean models. Moreover, we show that even
if the additional assumptions do not hold, this optimal estimator is
doubly-robust: it is consistent and asymptotically normal not only if the model
for treatment initiation is correct, but also if a certain outcome-regression
model is correct.
  We compare the optimal estimator to some naive choices within the class of
coarse structural nested mean models in a simulation study. Furthermore, we
apply the optimal and naive estimators to study how the CD4 count increase due
to one year of antiretroviral treatment (ART) depends on the time between HIV
infection and ART initiation in recently infected HIV infected patients. Both
in the simulation study and in the application, the use of optimal estimators
leads to substantial increases in precision.
