On the multiply robust estimation of the mean of the g-functional

Abstract: We study multiply robust (MR) estimators of the longitudinal g-computation formula of Robins (1986). In the first part of this paper we review and extend the recently proposed parametric multiply robust estimators of Tchetgen-Tchetgen (2009) and Molina, Rotnitzky, Sued and Robins (2017). In the second part of the paper we derive multiply and doubly robust estimators that use non-parametric machine-learning (ML) estimators of nuisance functions in lieu of parametric models. We use sample splitting to avoid the need for Donsker conditions, thereby allowing an analyst to select the ML algorithms of their choosing. We contrast the asymptotic behavior of our non-parametric doubly robust and multiply robust estimators. In particular, we derive formulas for their asymptotic bias. Examining these formulas we conclude that although, under certain data generating laws, the rate at which the bias of the MR estimator converges to zero can exceed that of the DR estimator, nonetheless, under most laws, the bias of the DR and MR estimators converge to zero at the same rate.