samples/texts/2704143/page_2.md

Strategy 3: The last strategy is based on a third representation of the M- functional

$${\iint }_{S\times X}\mathbb{E}(Y\mathrm{\mid }E=1,M=m,X=x)d{F}_{M\mathrm{\mid }E}(m\mathrm{\mid }E=0,X=x)d{F}_X(x)=\sum _{e=0}^1{\iiint }_{Y\times S\times X}y\frac{I(e=1)}{f_{E\mathrm{\mid }X}(e\mathrm{\mid }X=x)}\frac{f_{M\mathrm{\mid }E,X}(M\mathrm{\mid }E=0,X)}{f_{M\mathrm{\mid }E,X}(M\mathrm{\mid }E,X)}d{F}_{Y,M,E,X}(y,m,e,x)=\mathbb{E}\left\{Y\frac{I(E=1)}{f_{E\mathrm{\mid }X}(E\mathrm{\mid }X)}\frac{f_{M\mathrm{\mid }E,X}(M\mathrm{\mid }E=0,X)}{f_{M\mathrm{\mid }E,X}(M\mathrm{\mid }E,X)}\right\}\mathrm{.}\backslash iint\_\{S\; \backslash times\; X\}\; \backslash mathbb\{E\}(Y|E=1,\; M=m,\; X=x)\; dF\_\{M|E\}(m|E=0,\; X=x)\; dF\_X(x)\; \backslash \backslash \; =\; \backslash sum\_\{e=0\}^\{1\}\; \backslash iiint\_\{Y\; \backslash times\; S\; \backslash times\; X\}\; y\; \backslash frac\{I(e=1)\}\{f\_\{E|X\}(e|X=x)\}\; \backslash frac\{f\_\{M|E,X\}(M|E=0,\; X)\}\{f\_\{M|E,X\}(M|E,\; X)\}\; dF\_\{Y,M,E,X\}(y,\; m,\; e,\; x)\; \backslash \backslash \; =\; \backslash mathbb\{E\}\backslash left\backslash \{Y\; \backslash frac\{I(E=1)\}\{f\_\{E|X\}(E|X)\}\; \backslash frac\{f\_\{M|E,X\}(M|E=0,\; X)\}\{f\_\{M|E,X\}(M|E,\; X)\}\backslash right\backslash \}.$$

Thus, our third estimator takes the form

$${\widehat{\theta }}_0^{\text{em}}={\mathbb{P}}_n\left\{Y\frac{I(E=1)}{{\widehat{f}}_{E\mathrm{\mid }X}(E\mathrm{\mid }X)}\frac{{\widehat{f}}_{M\mathrm{\mid }E,X}(M\mathrm{\mid }E=0,X)}{{\widehat{f}}_{M\mathrm{\mid }E,X}(M\mathrm{\mid }E,X)}\right\}\mathrm{.}\backslash hat\{\backslash theta\}\_0^\{\backslash text\{em\}\}\; =\; \backslash mathbb\{P\}\_n\; \backslash left\backslash \{\; Y\; \backslash frac\{I(E=1)\}\{\backslash hat\{f\}\_\{E|X\}(E|X)\}\; \backslash frac\{\backslash hat\{f\}\_\{M|E,X\}(M|E=0,\; X)\}\{\backslash hat\{f\}\_\{M|E,X\}(M|E,\; X)\}\; \backslash right\backslash \}.$$

At first glance the three estimators $\hat{\theta}_0^{\text{em}}$, $\hat{\theta}0^{\text{ye}}$ and $\hat{\theta}0^{\text{ym}}$ might appear to be distinct; however, we observe that provided the empirical distribution function $\hat{F}O = \hat{F}{Y|E,M,X} \times \hat{F}{M|E,X} \times \hat{F}{E|X} \times \hat{F}_X$ satisfies the positivity assumption, and thus $\hat{F}O \in \mathcal{M}{\text{nonpar}}$, then actually $\hat{\theta}_0^{\text{em}} = \hat{\theta}0^{\text{ye}} = \hat{\theta}0^{\text{ym}} = \theta_0(\hat{F}O)$ since the three representations agree on the nonparametric model $\mathcal{M}{\text{nonpar}}$. Therefore we may conclude that these three estimators are in fact asymptotically efficient in $\mathcal{M}{\text{nonpar}}$ with common influence function $S{\theta_0}^{\text{eff,nonpar}}(\theta_0)$. Furthermore, from this observation, one further concludes that (asymptotic) inferences obtained using one of the three representations are identical to inferences using either of the other two representations.

At this juncture, we note that the above equivalence no longer applies when as we have previously argued will likely occur in practice, (M, X) contains 3 or more continuous variables and/or X is too high dimensional for models to be satu- rated or nonparametric, and thus parametric (or semiparametric) models are spec- ified for dimension reduction. Specifically, for such settings, we observe that three distinct modeling strategies are available. Under the first strategy, the estimator $\hat{\theta}_0^{\text{ym,par}}$ is obtained $\hat{\theta}0^{\text{ym}}$ using parametric model estimates $\hat{\mathbb{E}}^{\text{par}}(Y|E, M, X)$ and $\hat{f}{M|E,X}^{\text{par}}(m|E, X)$ instead of their nonparametric counterparts; similarly under the second strategy, the estimator $\hat{\theta}_0^{\text{ye,par}}$ is obtained similarly to $\hat{\theta}0^{\text{ye}}$ using estimates of parametric models $\hat{\mathbb{E}}^{\text{par}}(Y|E=1, M=m, X)$ and $\hat{f}{E|X}^{\text{par}}(e|X)$ and finally, un- der the third strategy, $\hat{\theta}0^{\text{em,par}}$ is obtained similarly to $\hat{\theta}0^{\text{em}}$ using $\hat{f}{E|X}^{\text{par}}(e|X)$ and $\hat{f}{M|E,X}^{\text{par}}(m|E, X)$. Then, it follows that $\hat{\theta}_0^{\text{ym,par}}$ is CAN under the submodel $\mathcal{M}a$, but is generally inconsistent if either $\hat{\mathbb{E}}^{\text{par}}(Y|E, M, X)$ or $\hat{f}{M|E,X}^{\text{par}}(m|E, X)$ fails to be consistent. Similarly, $\hat{\theta}_0^{\text{ye,par}}$ and $\hat{\theta}_0^{\text{em,par}}$ are, respectively, CAN under the submodels $\mathcal{M}_b$ and $\mathcal{M}_c$, but each estimator generally fails to be consistent out- side of the corresponding submodel. In the next section, we propose an approach