finite sample performance. But as recently shown by Kang and Schafer (2007) in the context of missing outcome data, a practical violation of positivity in data analysis can severely compromise inferences based on such methodology; although their analysis did not directly concern the M-functional $\theta_0$. Thus, it is crucial to critically examine, as we do below in a simulation study, the extent to which the various estimators discussed in this paper are susceptible to a practical violation of the positivity assumption, and to consider possible approaches to improve the finite sample performance of these estimators in the context of highly variable empirical weights. Methodology to enhance the finite sample behavior of $\hat{\delta}_j^{\text{doubly}}$ is well studied in the literature and is not considered here; see, for example, Robins et al. (2007), Cao, Tsiatis and Davidian (2009) and Tan (2010). We first describe an approach to enhance the finite sample performance of $\hat{\theta}_0^{\text{triple}}$, particularly in the presence of highly variable empirical weights. To focus the exposition, we only consider the case of a continuous $Y$ and a binary $M$, but in principle, the approach could be generalized to a more general setting. The proposed enhancement involves two modifications.
The first modification adapts to the mediation context, an approach developed for the missing data context (and for the estimation of total effects) in Robins et al. (2007). The basic guiding principle of the approach is to carefully modify the estimation of the outcome and mediator models in order to ensure that the triply robust estimator given by equation (4) has the simple M-functional representation
where $\hat{\eta}^{\text{par},\dagger}(1,0,X)$ is carefully estimated to ensure multiple robustness. The reason for favoring an estimator with the above representation is that it is expected to be more robust to practical positivity violation because it does not directly depend on inverse probability weights. However, as we show next, to ensure multiple robustness, estimation of $\eta^{\text{par}}$ involves inverse probability weights, and therefore, $\hat{\theta}_0^{\text{triple},\dagger}$ indirectly depends on such weights. Our strategy involves a second step to minimize the potential impact of this indirect dependence on weights.
In the following, we assume, to simplify the exposition, that a simple linear model is used:
Then, similar to Robins et al. (2007), one can verify that the above M-functional representation of a triply robust estimator is obtained by estimating $f_{M|E,X}^{\text{par}}(M|E=0, X)$ with $\hat{f}{M|E,X}^{\text{par},\dagger}(M|E=0, X)$ obtained via weighted logistic regression in the unexposed-only, with weight $\hat{f}{E|X}^{\text{par}}(0|X)^{-1}$; and by estimating $\mathbb{E}^{\text{par}}(Y|X, M, E=1)$ using weighted OLS of $Y$ on $(M, X)$ in the exposed-only, with weight