  Consider estimating the G-formula for the counterfactual mean outcome under a
given treatment regime in a longitudinal study. Bang and Robins provided an
estimator for this quantity that relies on a sequential regression formulation
of this parameter. This approach is doubly robust in that it is consistent if
either the outcome regressions or the treatment mechanisms are consistently
estimated. We define a stronger notion of double robustness, termed sequential
double robustness, for estimators of the longitudinal G-formula. The definition
emerges naturally from a more general definition of sequential double
robustness for the outcome regression estimators. An outcome regression
estimator is sequentially doubly robust (SDR) if, at each subsequent time
point, either the outcome regression or the treatment mechanism is consistently
estimated. This form of robustness is exactly what one would anticipate is
attainable by studying the remainder term of a first-order expansion of the
G-formula parameter. We show that a particular implementation of an existing
procedure is SDR. We also introduce a novel SDR estimator, whose development
involves a novel translation of ideas used in targeted minimum loss-based
estimation to the infinite-dimensional setting.
