SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \mu_ {a, T} = E \left\{Y _ {i} (a) \mid R _ {i} = T \right\} \\ = E \left[ E \left\{Y _ {i} (a) \mid V _ {i} = v, R _ {i} = T \right\} \mid R _ {i} = T \right] \\ = E \left[ E \left\{Y _ {i} (a) \mid V _ {i} = v, R _ {i} = k \right\} \mid R _ {i} = T \right] \\ = E \left(E \left[ E \left\{Y _ {i} (a) \mid X _ {i} = x, V _ {i} = v, R _ {i} = k \right\} \mid V _ {i} = v, R _ {i} = k \right] \mid R _ {i} = T\right) \\ = E \left(E \left[ E \left\{Y _ {i} (a) \mid X _ {i} = x, R _ {i} = k \right\} \mid V _ {i} = v, R _ {i} = k \right] \mid R _ {i} = T\right) \\ = E \left(E \left[ E \left\{Y _ {i} (a) \mid X _ {i} = x, R _ {i} = k, A _ {i} = a \right\} \mid V _ {i} = v, R _ {i} = k \right] \mid R _ {i} = T\right) \\ = E \left(E \left[ E \left\{Y _ {i} \mid X _ {i} = x, R _ {i} = k, A _ {i} = a \right\} \mid V _ {i} = v, R _ {i} = k \right] \mid R _ {i} = T\right). \tag {25} \\ \end{array}
$$
The second line follows the law of total expectation; the third line follows Assumption (A6); the fourth line follows the law of total expectation; the fifth line follows by our setup that  $V_{i} \subseteq X_{i}$ ; the sixth line follows Assumption (A4) the last line follows Assumption (A1).
E.2 Lemmas in [11]
We restate Lemmas E.1 and E.2 that were proved in [11]. These two lemmas together show that the  $L_{2}$  risks of the multiply robust estimators for  $\pi_{a,k}$  and  $m_{a,k}$  are bounded by the  $L_{2}$  risks of the model with the smallest risks, plus a negligible remainder term.
Lemma E.1. Suppose that for each  $j \in \mathcal{J}$ , there exists a constant  $0 < \epsilon_{j} < 1/2$  such that  $\epsilon_{j} < \widehat{\pi}_{a,k}^{j}(x) < 1 - \epsilon_{j}$  for all  $x$  and some  $k \in \mathcal{K}$ . Then
$$
E \left(\left\| \widehat {\pi} _ {a, k} - \pi_ {a, k} \right\| ^ {2}\right) \leq \inf  _ {j \in \mathcal {I}} \frac {2}{\epsilon_ {j} ^ {2}} E \left(\left\| \widehat {\pi} _ {a, k} ^ {j} - \pi_ {a, k} \right\| ^ {2}\right) + \frac {2 \log (J)}{n _ {k} - n _ {k} ^ {\text {t r a i n}}} \tag {26}
$$
where  $\widehat{\pi}_{a,k}(x) = \sum_{j=1}^{J} \widehat{\Lambda}_j \widehat{\pi}_{a,k}^j(x)$ .
If the candidate propensity models are parametric and one of them is correctly specified for  $\pi_{a,k}$ , then  $\inf_{j\in \mathcal{J}}\frac{2}{\epsilon_j^2} E(\|\widehat{\pi}_{a,k}^j -\pi_{a,k}\| ^2)$  converges at a rate of  $1 / n$ . If the candidate propensity models are non-parametric, then  $\inf_{j\in \mathcal{J}}\frac{2}{\epsilon_j^2} E(\|\widehat{\pi}_{a,k}^j -\pi_{a,k}\| ^2)$  converges at a rate slower than  $1 / n$ . The remainder term  $\frac{2\log(J)}{n_k - n_k^{\mathrm{train}}}$  converges at the rate  $1 / n_{k} - n_{k}^{\mathrm{train}}$ , so it vanishes at a faster rate than the statistical risks of candidate models themselves.
Lemma E.2. Suppose we have a continuous outcome and follow the continuous outcome model-mixing algorithm presented in Appendix B. Suppose there exist constants  $C_1, C_2 > 0$  such that  $\sup_{l \in \mathcal{L}} |\hat{m}_{a,k}^l(x) - m_{a,k}(x)| \leq C_1$  for all  $x$  and the subexponential norm of  $Y - m_{a,k}(x)$  given  $X = x$  is bounded above by  $C_2$  for all  $x$ . Then for  $a \in \{0,1\}$
$$
E \left(\| \widehat {m} _ {a, k} - m _ {a, k} \| ^ {2}\right) \leq \inf  _ {l \in \mathcal {L}} E \left(\| \widehat {m} _ {a, k} ^ {l} - m _ {a, k} \| ^ {2}\right) + \frac {\log (L)}{\kappa \left(n _ {k , a} - n _ {k , a} ^ {\text {t r a i n}}\right)}
$$
$$
0 <   \kappa \leq \max  \left[ \frac {1}{1 6 e C _ {1} C _ {2}}, \frac {\exp \left\{C _ {1} \left(8 e C _ {2}\right) ^ {- 1} \right\}}{4 \mathcal {M} _ {2} \left\{\left(4 e C _ {2}\right) ^ {- 1} \right\} + 1 6 C _ {1} ^ {2} \mathcal {M} _ {0} \left\{\left(4 e C _ {2}\right) ^ {- 1} \right\}} \right] \tag {27}
$$
where  $\mathcal{M}_0(t) = 2\exp \left(2e^2 C_2^2 t^2\right),\mathcal{M}_2(t) = 16\sqrt{2} C_2^2\exp \left(8e^4 C_2^2 t^2\right)$  and  $e = \exp (1)$ .
E.3 Influence function of site-specific estimator
In this section, we give the form of the influence functions for the site-specific estimators. The complete derivation is omitted here since it follows very closely to the derivation in [5] and [16]. The primary difference with the derivation in [16] is that we use the density ratio weights  $\zeta_k(V)$  instead of the inverse probability of selection weights. Define  $Z_{i} = (Y_{i},A_{i},X_{i},R_{i})$  with the (partial) baseline covariates in the target site as  $V_{i}\subseteq X_{i}$ . The general form for our efficient influence function is
$$
\begin{array}{l} \xi_ {a, k} (Z _ {i}) = \frac {1}{P (R _ {i} = k)} \left[ \frac {I (A _ {i} = a , R _ {i} = k)}{\pi_ {a , k} (X _ {i})} \zeta_ {k} (V _ {i}) \{Y _ {i} - m _ {a, k} (X _ {i}) \} \right] \\ + \frac {1}{P (R _ {i} = k)} \left[ I (R _ {i} = k) \zeta_ {k} (V _ {i}) \{m _ {a, k} (X _ {i}) - \tau_ {a, k} (V _ {i}) \} \right] \\ + \frac {1}{P \left(R _ {i} = T\right)} \left[ I \left(R _ {i} = T\right) \tau_ {a, k} \left(V _ {i}\right) \right] - \mu_ {a, T}. \tag {28} \\ \end{array}
$$
The target site estimator has the following form,
$$
\widehat {\mu} _ {a, T} = \frac {1}{n _ {T}} \sum_ {i = 1} ^ {n} \left[ \frac {I \left(A _ {i} = a , R _ {i} = T\right)}{\pi_ {a , T} (V _ {i})} \left\{Y _ {i} - m _ {a, T} (V _ {i}) \right\} + I (R _ {i} = T) m _ {a, T} (V _ {i}) \right]. \tag {29}
$$
This is a standard AIPW estimator whose influence function is derived in [2, 6] as
$$
\xi_ {a, T} \left(Z _ {i}\right) = \frac {1}{P \left(R _ {i} = T\right)} \left[ \frac {I \left(A _ {i} = a , R _ {i} = T\right)}{\pi_ {a , T} \left(V _ {i}\right)} \left\{Y _ {i} - m _ {a, T} \left(V _ {i}\right) \right\} + I \left(R _ {i} = T\right) m _ {a, T} \left(V _ {i}\right) \right] - \mu_ {a, T}. \tag {30}
$$
E.4 Proof of Theorem 2
We first prove that given the conditions in Theorem 2, the site-specific estimator is a consistent estimator of  $\mu_{a,T}$ , formalized in the following Lemma:
Lemma E.3. Given Assumptions (A1) - (A7), and E.1 and E.2, if one of (B1) or (B2) and one of (C1) or (C2) hold, then  $\widehat{\mu}_{a,k}$  is a consistent estimator for  $\mu_{a,T}$ .
Proof of Lemma E.3. We divide the proof into four cases and show that  $\widehat{\mu}_{a,k}$  achieves consistency.
Case 1: When  $\overline{\pi}_{a,k}^{j} = \pi_{a,k}$  and  $\overline{\tau}_{a,k} = \tau_{a,k}$
By assumption, we have  $\| \widehat{\pi}_{a,k}^{j} - \pi_{a,k} \| = o_{p}(1)$  and  $\| \widehat{\tau}_{a,k} - \tau_{a,k} \| = o_{p}(1)$ . Given Lemma E.1 for  $\widehat{\pi}_{a,k}$ ,  $\| \widehat{\pi}_{a,k} - \pi_{a,k} \| = o_{p}(1)$ . Define  $\overline{m}_{a,k}(x) = \frac{1}{L} \sum_{l=1}^{L} \overline{m}_{a,k}^{l}(x)$ . By definition,  $\| \widehat{m}_{a,k}^{l} - \overline{m}_{a,k}^{l} \| = o_{p}(1)$ , so  $\| \widehat{m}_{a,k} - \overline{m}_{a,k} \| = o_{p}(1)$ . Together with  $\| \widehat{\zeta}_{k} - \overline{\zeta}_{k} \| = o_{p}(1)$ , we can re-write the site-specific estimator as
$$
\begin{array}{l} \widehat {\mu} _ {a, k} = \frac {1}{n} \sum_ {i = 1} ^ {n} \left[ \frac {n}{n _ {k}} \frac {I (A _ {i} = a , R _ {i} = k)}{\pi_ {a , k} (X _ {i})} \overline {{\zeta}} _ {k} (V _ {i}) \Big \{Y _ {i} - \overline {{m}} _ {a, k} (X _ {i}) \Big \} \right] \\ + \frac {1}{n} \sum_ {i = 1} ^ {n} \Big [ \frac {n}{n _ {k}} I (R _ {i} = k) \overline {{\zeta}} _ {k} (V _ {i}) \Big \{\overline {{m}} _ {a, k} (X _ {i}) - \tau_ {a, k} (V _ {i}) \Big \} \Big ] \\ + \frac {1}{n} \sum_ {i = 1} ^ {n} \frac {n}{n _ {T}} I (R _ {i} = T) \tau_ {a, k} (V _ {i}) \\ + o _ {p} (1) \tag {31} \\ \end{array}
$$
By assumption of i.i.d. units within each site and the law of large numbers,  $\widehat{\mu}_{a,k}$  converges in probability to
$$
\begin{array}{l} E \left[ \frac {n}{n _ {k}} \frac {I (A _ {i} = a , R _ {i} = k)}{\pi_ {a , k} (X _ {i})} \bar {\zeta} _ {k} (V _ {i}) \left\{Y _ {i} - \bar {m} _ {a, k} (X _ {i}) \right\} \right] \\ + E \Bigg [ \frac {n}{n _ {k}} I (R _ {i} = k) \overline {{\zeta}} _ {k} (V _ {i}) \Big \{\overline {{m}} _ {a, k} (X _ {i}) - \tau_ {a, k} (V _ {i}) \Big \} \Bigg ] \\ + E \left[ \frac {n}{n _ {T}} I \left(R _ {i} = T\right) \tau_ {a, k} \left(V _ {i}\right) \right] \\ = \underbrace {E \left[ \frac {n}{n _ {k}} I (R _ {i} = k) \bar {\zeta} _ {k} (V _ {i}) \left\{\frac {I (A _ {i} = a)}{\pi_ {a , k} (X _ {i})} Y _ {i} - \tau_ {a , k} (V _ {i}) \right\} \right]} _ {T _ {1}} \\ - \underbrace {E \left[ \frac {n}{n _ {k}} I (R _ {i} = k) \bar {\zeta} _ {k} (V _ {i}) \left\{\frac {I (A _ {i} = a)}{\pi_ {a , k} (X _ {i})} - 1 \right\} \bar {m} _ {a , k} (X _ {i}) \right]} _ {T _ {2}} \\ + \underbrace {E \left[ \frac {n}{n _ {T}} I \left(R _ {i} = T\right) \tau_ {a , k} \left(V _ {i}\right) \right]} _ {T _ {3}}. \tag {32} \\ \end{array}
$$
Given that  $\pi_{a,k}$  is the true propensity score model,  $E(T_2) = 0$  since
$$
E \left\{\frac {I \left(A _ {i} = a\right)}{\pi_ {a , k} \left(X _ {i}\right)} - 1 \mid X _ {i}, R _ {i} = k \right\} = \frac {P \left(A _ {i} = a \mid X _ {i} , R _ {i} = k\right)}{\pi_ {a , k} \left(X _ {i}\right)} - 1 = 0. \tag {33}
$$
Similarly, given that  $\tau_{a,k} = E\{m_{a,k}(X_i) \mid V_i, R_i = k\}$  is the true model,
$$
\begin{array}{l} E \left(T _ {3}\right) = E \left\{E \left(T _ {3} \mid R _ {i} = T\right) \right\} \\ = \frac {n}{n _ {T}} P (R _ {i} = T) E (\tau_ {a, k} (V _ {i}) \mid R _ {i} = T) \\ = E \left(\tau_ {a, k} \left(V _ {i}\right) \mid R _ {i} = T\right) \\ = \mu_ {a, T}. \tag {34} \\ \end{array}
$$
Finally, we consider  $T_{1}$
$$
\begin{array}{l} E (T _ {1}) = E \Bigg [ \frac {n}{n _ {k}} I (R _ {i} = k) \overline {{\zeta}} _ {k} (V _ {i}) \Big \{\frac {I (A _ {i} = a)}{\pi_ {a , k} (X _ {i})} Y _ {i} - \tau_ {a, k} (V _ {i}) \Big \} \Bigg ] \\ = E \left(E \left[ \frac {n}{n _ {k}} I (R _ {i} = k) \bar {\zeta} _ {k} \left(V _ {i}\right) \left\{\frac {I \left(A _ {i} = a\right)}{\pi_ {a , k} \left(X _ {i}\right)} Y _ {i} - \tau_ {a, k} \left(V _ {i}\right) \right\} \mid V _ {i} \right]\right) \\ = E \left(\frac {n}{n _ {k}} I (R _ {i} = k) \bar {\zeta} _ {k} (V _ {i}) \left[ E \left\{\frac {I (A _ {i} = a)}{\pi_ {a , k} (X _ {i})} Y _ {i} \mid V _ {i} \right\} - \tau_ {a, k} (V _ {i}) \right]\right) \\ = E \left(\frac {n}{n _ {k}} I (R _ {i} = k) \bar {\zeta} _ {k} (V _ {i}) \left[ E \left\{\frac {I (A _ {i} = a)}{\pi_ {a , k} (X _ {i})} Y _ {i} \mid V _ {i}, R _ {i} = k \right\} - \tau_ {a, k} (V _ {i}) \right]\right) \\ = E \left\{\frac {n}{n _ {k}} I (R _ {i} = k) \bar {\zeta} _ {k} (V _ {i}) \left(E \left[ E \left\{\frac {I (A _ {i} = a)}{\pi_ {a , k} (X _ {i})} Y _ {i} \mid X _ {i}, R _ {i} = k \right\} \mid V _ {i}, R _ {i} = k \right] - \tau_ {a, k} (V _ {i})\right) \right\} \\ = E \left(\frac {n}{n _ {k}} I \left(R _ {i} = k\right) \bar {\zeta} _ {k} \left(V _ {i}\right) \left[ E \left\{E \left(Y _ {i} \mid A _ {i} = a, X _ {i}, R _ {i} = k\right) \mid V _ {i}, R _ {i} = k \right\} - \tau_ {a, k} \left(V _ {i}\right) \right] \right. \\ = E \left(\frac {n}{n _ {k}} I \left(R _ {i} = k\right) \bar {\zeta} _ {k} \left(V _ {i}\right) \left[ E \left\{m _ {a, k} \left(X _ {i}\right) \mid V _ {i}, R _ {i} = k \right\} - \tau_ {a, k} \left(V _ {i}\right) \right]\right) \\ = 0, \text {w h i c h c o m p l e t e s t h e} \tag {35} \\ \end{array}
$$
Case 2: When  $\overline{\pi}_{a,k}^{j} = \pi_{a,k}$  and  $\overline{\zeta}_k = \zeta_k$
By assumption, we have  $\| \widehat{\pi}_{a,k}^{j} - \pi_{a,k} \| = o_{p}(1)$  and  $\| \widehat{\zeta}_{k} - \zeta_{k} \| = o_{p}(1)$ . Given Lemma E.1 for  $\widehat{\pi}_{a,k}$ ,  $\| \widehat{\pi}_{a,k} - \pi_{a,k} \| = o_{p}(1)$ . Define  $\overline{m}_{a,k}(x) = \frac{1}{L} \sum_{l=1}^{L} \overline{m}_{a,k}^{l}(x)$ . By definition,  $\| \widehat{m}_{a,k}^{l} - \overline{m}_{a,k}^{l} \| = o_{p}(1)$ , so  $\| \widehat{m}_{a,k} - \overline{m}_{a,k} \| = o_{p}(1)$ . Together with  $\| \widehat{\tau}_{a,k} - \overline{\tau}_{a,k} \| = o_{p}(1)$ , we can write the site-specific estimator as
$$
\begin{array}{l} \widehat {\mu} _ {a, k} = \frac {1}{n} \sum_ {i = 1} ^ {n} \left[ \frac {n}{n _ {k}} \frac {I (A _ {i} = a , R _ {i} = k)}{\pi_ {a , k} (X _ {i})} \zeta_ {k} (V _ {i}) \Big \{Y _ {i} - \overline {{m}} _ {a, k} (X _ {i}) \Big \} \right] \\ + \frac {1}{n} \sum_ {i = 1} ^ {n} \left[ \frac {n}{n _ {k}} I (R _ {i} = k) \zeta_ {k} (V _ {i}) \left\{\bar {m} _ {a, k} (X _ {i}) - \bar {\tau} _ {a, k} (V _ {i}) \right\} \right] \\ + \frac {1}{n} \sum_ {i = 1} ^ {n} \frac {n}{n _ {T}} I \left(R _ {i} = T\right) \bar {\tau} _ {a, k} \left(V _ {i}\right) \\ + o _ {p} (1) \tag {36} \\ \end{array}
$$
$$
\begin{array}{l} E \left[ \frac {n}{n _ {k}} \frac {I (A _ {i} = a , R _ {i} = k)}{\pi_ {a , k} (X _ {i})} \zeta_ {k} (V _ {i}) \left\{Y _ {i} - \overline {{m}} _ {a, k} (X _ {i}) \right\} \right] \\ + E \left[ \frac {n}{n _ {k}} I (R _ {i} = k) \zeta_ {k} (V _ {i}) \left\{\overline {{m}} _ {a, k} (X _ {i}) - \overline {{\tau}} _ {a, k} (V _ {i}) \right\} \right] \\ + E \left[ \frac {n}{n _ {T}} I (R _ {i} = T) \bar {\tau} _ {a, k} (V _ {i}) \right] \\ \end{array}
$$
$$
\begin{array}{l} = \underbrace {E \left\{\frac {n}{n _ {k}} \frac {I (A _ {i} = a , R _ {i} = k)}{\pi_ {a , k} (X _ {i})} \zeta_ {k} (V _ {i}) Y _ {i} \right\}} _ {T _ {1}} \\ + \underbrace {E \left[ \frac {n}{n _ {k}} I (R _ {i} = k) \zeta_ {k} \left(V _ {i}\right) \bar {m} _ {a , k} \left(X _ {i}\right) \left\{1 - \frac {I \left(A _ {i} = a\right)}{\pi_ {a , k} \left(X _ {i}\right)} \right\} \right]} _ {T _ {2}} \\ + \underbrace {E \left[ \bar {\tau} _ {a , k} (V _ {i}) \left\{\frac {n}{n _ {T}} I (R _ {i} = T) - \frac {n}{n _ {k}} I (R _ {i} = k) \zeta_ {k} (V _ {i}) \right\} \right]} _ {T _ {3}} \tag {37} \\ \end{array}
$$
$$
E \left\{\frac {I \left(A _ {i} = a\right)}{\pi_ {a , k} \left(X _ {i}\right)} - 1 \mid X _ {i}, R _ {i} = k \right\} = \frac {P \left(A _ {i} = a \mid X _ {i} , R _ {i} = k\right)}{\pi_ {a , k} \left(X _ {i}\right)} - 1 = 0. \tag {38}
$$
Also since  $\zeta_{k} = f(V_{i}|R_{i} = T) / f(V_{i}|R_{i} = k)$  is the true density ratio model,
$$
\begin{array}{l} E \left(T _ {3}\right) = E \left(E \left[ \bar {\tau} _ {a, k} \left(V _ {i}\right) \left\{\frac {n}{n _ {T}} I \left(R _ {i} = T\right) - \frac {n}{n _ {k}} I \left(R _ {i} = k\right) \zeta_ {k} \left(V _ {i}\right) \right\} \mid V _ {i} \right]\right) \\ = E \left(\bar {\tau} _ {a, k} (V _ {i}) \left[ \frac {n}{n _ {T}} E \left\{I (R _ {i} = T) \right\} - \frac {n}{n _ {k}} E \left\{I (R _ {i} = k) \right\} \zeta_ {k} (V _ {i}) \mid V _ {i} \right]\right) \\ = 0 \text {f o l l o w i n g t h e r e s u l t s f r o m A p p e n d i x A .} \tag {39} \\ \end{array}
$$
Thus, we are left with  $T_{1}$ , which equals to  $\mu_{a,T}$  by iterative conditioning and Assumptions (A1), (A2), (A4) and (A6).
Proofs for Case 3, when  $\overline{m}_{a,k}^{l} = m_{a,k}$  and  $\overline{\tau}_{a,k} = \tau_{a,k}$  and Case 4, when  $\overline{m}_{a,k}^{l} = m_{a,k}$  and  $\overline{\zeta}_k = \zeta_k$  follow closely to the proofs of Cases 1 and 2 and are omitted here for brevity.
Proof of Theorem 2. Given the general form of influence function (28), we can decompose the estimation risk of the site-specific estimator as
$$
\mathbb {P} _ {n} \left(\widehat {\mu} _ {a, k}\right) - E \left(\mu_ {a, T}\right) = \left(\mathbb {P} _ {n} - E\right) \left(\widehat {\mu} _ {a, k} - \mu_ {a, T}\right) + \left(\mathbb {P} _ {n} - E\right) \mu_ {a, T} + E \left(\widehat {\mu} _ {a, k} - \mu_ {a, T}\right) \tag {40}
$$
Assuming proper sample splitting methods are employed, as stated in Lemma 2 of [9], we have:
$$
\left(\mathbb {P} _ {n} - E\right) \left(\widehat {\mu} _ {a, k} - \mu_ {a, T}\right) = o _ {p} \left(n ^ {- 1 / 2}\right). \tag {41}
$$
Then, we consider  $(\mathbb{P}_n - E)\mu_{a,T}$ . By the Central Limit Theorem, we have  $\sqrt{n}\left\{\mathbb{P}_n(\mu_{a,T}) - E(\mu_{a,T})\right\} \stackrel{d}{\to} \mathcal{N}(0,\sigma^2)$ , where  $\sigma^2$  is given by:
$$
\begin{array}{l} \sigma^ {2} = E \left\{\xi_ {a, k} (Z _ {i}) ^ {2} \right\} \\ = E \left\{\frac {1}{P ^ {2} (R _ {i} = k)} \left[ \frac {I (A _ {i} = a , R _ {i} = k)}{\pi_ {a , k} (X _ {i})} \zeta_ {k} (V _ {i}) \{Y _ {i} - m _ {a, k} (X _ {i}) \} \right] ^ {2} \right\} \\ + E \left\{\frac {1}{P ^ {2} (R _ {i} = k)} \left[ I (R _ {i} = k) \zeta_ {k} (V _ {i}) \left\{m _ {a, k} (X _ {i}) - \tau_ {a, k} (V _ {i}) \right\} \right] ^ {2} \right\} \\ + E \left\{\frac {1}{P ^ {2} (R _ {i} = T)} \left[ I (R _ {i} = T) \left\{\tau_ {a, k} (V _ {i}) - \mu_ {a, T} \right\} \right] ^ {2} \right\} \\ + \text {R e m a i n i n g c r o s s - t e r m s}. \tag {42} \\ \end{array}
$$
It can be verified that all remaining cross-terms have an expected value of zero. Hence,
$$
\sigma^ {2} = E \left\{\frac {1}{P ^ {2} (R _ {i} = k)} \left[ \frac {I (A _ {i} = a , R _ {i} = k)}{\pi_ {a , k} ^ {2} (X _ {i})} \zeta_ {k} ^ {2} (V _ {i}) \{Y _ {i} - m _ {a, k} (X _ {i}) \} ^ {2} \right] \right\}
$$
$$
\begin{array}{l} + E \left\{\frac {1}{P ^ {2} (R _ {i} = k)} I (R _ {i} = k) \left[ \zeta_ {k} \left(V _ {i}\right) \left\{m _ {a, k} \left(X _ {i}\right) - \tau_ {a, k} \left(V _ {i}\right) \right\} \right] ^ {2} \right\} \\ + E \left[ \frac {1}{P ^ {2} (R _ {i} = T)} I (R _ {i} = T) \left\{\tau_ {a, k} (V _ {i}) - \mu_ {a, T} \right\} ^ {2} \right] \\ = E \left[ E \left\{\frac {1}{P ^ {2} \left(R _ {i} = k\right)} \left[ \frac {I \left(A _ {i} = a , R _ {i} = k\right)}{\pi_ {a , k} ^ {2} \left(X _ {i}\right)} \zeta_ {k} ^ {2} \left(V _ {i}\right) \left\{Y _ {i} - m _ {a , k} \left(X _ {i}\right) \right\} ^ {2} \right] \mid X _ {i} \right\} \right] \\ + E \left[ E \left\{\frac {1}{P ^ {2} (R _ {i} = k)} I (R _ {i} = k) \zeta_ {k} ^ {2} (V _ {i}) \{m _ {a, k} (X _ {i}) - \tau_ {a, k} (V _ {i}) \} ^ {2} \mid V _ {i} \right\} \right] \\ + E \left[ \frac {1}{P ^ {2} (R _ {i} = T)} I (R _ {i} = T) \left\{\tau_ {a, k} (V _ {i}) - \mu_ {a, T} \right\} ^ {2} \right] \\ = E \left[ \frac {P (R _ {i} = k \mid X _ {i})}{P ^ {2} (R _ {i} = k)} \zeta_ {k} ^ {2} (V _ {i}) E \left\{\frac {I (A _ {i} = a)}{\pi_ {a , k} ^ {2} (X _ {i})} \{Y _ {i} - m _ {a, k} (X _ {i}) \} ^ {2} \mid X _ {i}, A _ {i} = a, R _ {i} = k \right\} \right] \\ + E \left\{\frac {P \left(R _ {i} = k \mid X _ {i}\right)}{P ^ {2} \left(R _ {i} = k\right)} \zeta_ {k} ^ {2} \left(V _ {i}\right) E \left[ \left\{m _ {a, k} \left(X _ {i}\right) - \tau_ {a, k} \left(V _ {i}\right) \right\} ^ {2} \mid X _ {i}, R _ {i} = k \right] \right\} \\ + E \left(\frac {P \left(R _ {i} = T\right)}{P ^ {2} \left(R _ {i} = T\right)} E \left[ \left\{\tau_ {a, k} \left(V _ {i}\right) - \mu_ {a, T} \right\} ^ {2} \mid R _ {i} = T \right]\right) \\ = E \left[ \frac {P \left(R _ {i} = k \mid V _ {i}\right)}{P ^ {2} \left(R _ {i} = k\right)} \zeta_ {k} ^ {2} \left(V _ {i}\right) \frac {\operatorname {V a r} \left\{Y _ {i} \mid X _ {i} , A _ {i} = a , R _ {i} = k \right\}}{\pi_ {a , k} \left(X _ {i}\right)} \right] \\ + E \left[ \frac {P \left(R _ {i} = k \mid V _ {i}\right)}{P ^ {2} \left(R _ {i} = k\right)} \zeta_ {k} ^ {2} \left(V _ {i}\right) \operatorname {V a r} \left\{m _ {a, k} \left(X _ {i}\right) \mid V _ {i}, R _ {i} = k \right\} \right] \\ + E \left[ \frac {\operatorname {V a r} \left\{\tau_ {a , k} \left(V _ {i}\right) \mid R _ {i} = T \right\}}{P \left(R _ {i} = T\right)} \right]. \tag {43} \\ \end{array}
$$
This expression represents the semiparametric efficiency bound for estimating  $\mu_{a,T}$  and is finite given that all nuisance functions are uniformly bounded. Consequently, we can conclude that  $(\mathbb{P}_n - E)(\mu_{a,T}) = O_p(n^{-1 / 2})$
Moving forward, we analyze the conditional bias term. Firstly, let us define  $E\{\hat{m}_{a,k}(X_i) \mid V_i, R_i = k\} = \tilde{\tau}_{a,k}(V_i)$ . Given conditions in Theorem 2 and Lemma E.3, we can rewrite the conditional bias as follows:
$$
\begin{array}{l} E \left(\widehat {\mu} _ {a, k} - \mu_ {a, T}\right) = E \left(\frac {n}{n _ {k}} \left[ \frac {I \left(A _ {i} = a , R _ {i} = k\right)}{\widehat {\pi} _ {a , k} \left(X _ {i}\right)} \widehat {\zeta} _ {k} \left(V _ {i}\right) \left\{m _ {a, k} \left(X _ {i}\right) - \widehat {m} _ {a, k} \left(X _ {i}\right) \right\} \right]\right) \\ + E \left(\frac {n}{n _ {k}} \left[ I (R _ {i} = k) \widehat {\zeta} _ {k} (V _ {i}) \left\{\tilde {\tau} _ {a, k} (V _ {i}) - \widehat {\tau} _ {a, k} (V _ {i}) \right\} \right]\right) \\ + E \left(\frac {n}{n _ {T}} \left[ I \left(R _ {i} = T\right) \left\{\widehat {\tau} _ {a, k} \left(V _ {i}\right) - \mu_ {a, T} \left(V _ {i}\right) \right\} \right]\right) \\ = E \left(\frac {n}{n _ {k}} \left[ \frac {I \left(A _ {i} = a , R _ {i} = k\right)}{\widehat {\pi} _ {a , k} \left(X _ {i}\right)} \widehat {\zeta} _ {k} \left(V _ {i}\right) \left\{m _ {a, k} \left(X _ {i}\right) - \widehat {m} _ {a, k} \left(X _ {i}\right) \right\} \right]\right) \\ + E \left(\frac {n}{n _ {k}} \left[ I (R _ {i} = k) \widehat {\zeta} _ {k} (V _ {i}) \left\{\tilde {\tau} _ {a, k} (V _ {i}) - \tau_ {a, k} (V _ {i}) \right\} \right]\right) \\ \left. \right. - E \left(\frac {n}{n _ {k}} \left[ I \left(R _ {i} = k\right) \widehat {\zeta} _ {k} \left(V _ {i}\right)\left\{\widehat {\tau} _ {a, k} \left(V _ {i}\right) - \tau_ {a, k} \left(V _ {i}\right)\right\}\right]\right) \\ + E \left(\frac {n}{n _ {T}} \left[ I \left(R _ {i} = T\right) \left\{\widehat {\tau} _ {a, k} \left(V _ {i}\right) - \mu_ {a, T} \left(V _ {i}\right) \right\} \right]\right) \\ = E \left(\frac {n}{n _ {k}} \left[ \frac {I \left(A _ {i} = a , R _ {i} = k\right)}{\widehat {\pi} _ {a , k} (X _ {i})} \widehat {\zeta} _ {k} (V _ {i}) \left\{m _ {a, k} (X _ {i}) - \widehat {m} _ {a, k} (X _ {i}) \right\} \right]\right) \\ + E \left(\frac {n}{n _ {k}} \left[ I \left(R _ {i} = k\right) \widehat {\zeta} _ {k} \left(V _ {i}\right) \left\{\widehat {m} _ {a, k} \left(X _ {i}\right) - m _ {a, k} \left(X _ {i}\right) \right\} \right]\right) \\ \end{array}
$$
$$
\begin{array}{l} \left. - E \left(\frac {n}{n _ {k}} \left[ I \left(R _ {i} = k\right) \widehat {\zeta} _ {k} \left(V _ {i}\right) \left\{\widehat {\tau} _ {a, k} \left(V _ {i}\right) - \tau_ {a, k} \left(V _ {i}\right) \right\} \right]\right)\right) \\ + E \left(\frac {n}{n _ {T}} \left[ I \left(R _ {i} = T\right) \left\{\widehat {\tau} _ {a, k} \left(V _ {i}\right) - \mu_ {a, T} \left(V _ {i}\right) \right\} \right]\right) \\ = \underbrace {E \left[ \frac {n}{n _ {k}} \widehat {\zeta} _ {k} (V _ {i}) \left\{\frac {I \left(A _ {i} = a , R _ {i} = k\right)}{\widehat {\pi} _ {a , k} (X _ {i})} - I \left(R _ {i} = k\right) \right\} \left\{m _ {a , k} \left(X _ {i}\right) - \widehat {m} _ {a , k} \left(X _ {i}\right) \right\} \right]} _ {T _ {1}} \\ + \underbrace {E \left[ \left\{\frac {n}{n _ {T}} I (R _ {i} = T) - \frac {n}{n _ {k}} I (R _ {i} = k) \widehat {\zeta} _ {k} (V _ {i}) \right\} \left\{\widehat {\tau} _ {a , k} (V _ {i}) - \tau_ {a , T} (V _ {i}) \right\} \right]} _ {T _ {2}}. \tag {44} \\ \end{array}
$$
Step 3 is derived from Step 2 through conditioning on the variables  $(V_{i},R_{i} = k)$ , and we obtain the following equality:
$$
\left. E \left(\frac {n}{n _ {k}} \left[ I \left(R _ {i} = k\right) \widehat {\zeta} _ {k} \left(V _ {i}\right) \left\{\tilde {\tau} _ {a, k} \left(V _ {i}\right) - \tau_ {a, k} \left(V _ {i}\right) \right\} \right]\right) = E \left(\frac {n}{n _ {k}} \left[ I \left(R _ {i} = k\right) \widehat {\zeta} _ {k} \left(V _ {i}\right) \left\{\widehat {m} _ {a, k} \left(V _ {i}\right) - m _ {a, k} \left(V _ {i}\right) \right\} \right]\right)\right). \tag {45}
$$
Step 4 can be derived from Step 3 by definition; in the target site, the conditional outcome is represented by  $\mu_{a,T} = m_{a,T}(V_i)$  and  $\tau_{a,T} = E\{m_{a,T}(V_i) \mid V_i, R_i = T\} = m_{a,T}(V_i) = \mu_{a,T}$ . Conditioning on  $X_i$ , we have the following expression: