# C More results and their proofs

## C.1 Estimation of the asymptotic variance of an estimator

### C.1.1 IPTW estimator based on a well-specified model

**Sketch of proof.** The IPTW estimator \(\psi_{n}^{b}\) relies on algorithm
\(\Algo_{\Gbar,1}\), which is “well-specified” in the
sense that its output \(\Gbar_{n}\defq \Algo_{\Gbar,1}(P_{n})\) minimizes the
empirical risk over a finite-dimensional, identifiable, well-specified working
model for \(\Gbar_{0}\). If one introduces \(D\) given by

\[\begin{equation*} D(O) \defq \frac{(2A-1)}{\ell\Gbar_{0}(A,W)} Y, \end{equation*}\]

then the influence curve of \(\psi_{n}^{b}\) equals \(D - \Psi(P_{0})\) minus the projection of \(D\) onto the tangent space of the above parametric model for \(\Gbar_{0}\). The variance of the influence curve is thus smaller than that of \(D\), hence the conservativeness.

### C.1.2 G-computation estimator based on a well-specified model

**Sketch of proof** (see (Laan and Rose 2011) page 527). Consider a G-computation
estimator \(\psi_{n}\) that relies on an algorithm \(\Algo_{\Qbar}\) that is
“well-specified” in the sense that its output
\(\Qbar_{n}\defq \Algo_{\Qbar}(P_{n})\) minimizes the empirical risk over a
finite-dimensional, identifiable, well-specified working model for
\(\Qbar_{0}\). If one introduces \(D\) given by

\[\begin{equation*} D(O) \defq \Qbar_{0}(1,W) - \Qbar_{0}(0,W) \end{equation*}\]

then the influence curve of \(\psi_{n}\) equals \(D - \Psi(P_{0})\) plus a function of \(O\) that is orthogonal to \(D - \Psi(P_{0})\). Thus the variance of the influence curve is larger than that of \(D\), hence the anti-conservativeness.

## C.2 ☡ General analysis of plug-in estimators

Recall that \(\Algo_{Q_{W}}\) is an algorithm designed for the estimation of \(Q_{0,W}\) (see Section 7.3) and that we denote by \(Q_{n,W} \defq \Algo_{Q_{W}}(P_{n})\) the output of the algorithm trained on \(P_{n}\). Likewise, \(\Algo_{\Gbar}\) and \(\Algo_{\Qbar}\) are two generic algorithms designed for the estimation of \(\Gbar_{0}\) and of \(\Qbar_{0}\) (see Sections 7.4 and 7.6), \(\Gbar_{n} \defq \Algo_{\Gbar}(P_{n})\) and \(\Qbar_{n} \defq \Algo_{\Qbar}(P_{n})\) are their outputs once trained on \(P_{n}\).

Let us now introduce \(\Phat_n\) a law in \(\calM\) such that the \(Q_{W}\), \(\Gbar\)
and \(\Qbar\) features of \(\Phat_n\) equal \(Q_{n,W}\), \(\Gbar_{n}\) and
\(\Qbar_{n}\), respectively. We say that any such law is *compatible* with
\(Q_{n,W}\), \(\Gbar_n\) and \(\Qbar_n\).

### C.2.1 Main analysis

Substituting \(\Phat_n\) for \(P\) in (4.1) yields (9.1): \[\begin{equation} \sqrt{n} (\Psi(\Phat_n) - \Psi(P_0)) = - \sqrt{n} P_0 D^*(\Phat_n) + \sqrt{n} \Rem_{P_0}(\Phat_n), \end{equation}\]

an equality that we rewrite as \[\begin{align} \sqrt{n} (\Psi(\Phat_n) - \Psi(P_0)) = - & \sqrt{n} P_n D^*(\Phat_n) + \sqrt{n} (P_n - P_0) D^*(P_0)\\ & + \sqrt{n}(P_n - P_0) [D^*(\Phat_n) - D^*(P_0)] + \sqrt{n}\Rem_{P_0}(\Phat_n). \end{align}\]

Let us know study in turn the four terms in the above right-hand side sum. Recall that \(X_n = o_{P_0}(1)\) means that \(P_0(|X_n| > t)\) converges to zero for all \(t>0\) as \(n\) goes to infinity.

In view of (4.4), the fourth term is \(o_{P_0}(1)\) provided that \(\sqrt{n}\|\Qbar - \Qbar_{0}\|_{P_0} \times \|(\Gbar - \Gbar_{0})/\ell\Gbar_{0}\|_{P_0} = o_{P_0}(1)\). This is the case if, for instance, \(\ell\Gbar_{0}\) is bounded away from zero, and both \(n^{1/4}\|\Qbar - \Qbar_{0}\|_{P_0}\) and \(n^{1/4}\|\Gbar - \Gbar_{0}\|_{P_0}\) are \(o_{P_0}(1)\). What really matters, remarkably, is the

*product*of the two norms. If each norm goes to zero at rate \(n^{1/4}\), then their product does at rate \(\sqrt{n}\). Of course, if one goes to zero at rate \(n^{1/4 + c}\) for some \(0<c<1/4\), then it suffices that the other go to zero at rate \(n^{1/4 - c}\). See also Section C.3.A fundamental result from empirical processes theory gives us conditions guaranteeing that the third term is \(o_{P_0}(1)\). By Lemma 19.24 in (Vaart 1998), this is the case indeed if \(\|D^*(\Phat_n) - D^*(P_0)\|_{P_{0}} = o_{P_0} (1)\) (that is, if \(D^*(\Phat_n)\) estimates consistently \(D^*(P_0)\)) and if \(D^*(\Phat_n)\) falls (with probability tending to one) into a Donsker class (meaning that the random \(D^*(\Phat_n)\) must belong eventually to a set that is not too large). Requesting that \(\|D^*(\Phat_n) - D^*(P_0)\|_{P_{0}} = o_{P_0} (1)\) is not much if one is already willing to assume that \(n^{1/4}\|\Qbar - \Qbar_{0}\|_{P_0}\) and \(n^{1/4}\|\Gbar - \Gbar_{0}\|_{P_0}\) are \(o_{P_0}(1)\). Moreover, the second condition can be interpreted as a condition on the complexity/versatility of algorithms \(\Algo_{\Gbar}\) and \(\Algo_{\Qbar}\).

By the central limit theorem, the second term converges in law to the centered Gaussian law with variance \(P_0 D^*(P_0)^2\).

As for the first term, all we can say is that it is a potentially large (because of the \(\sqrt{n}\) renormalization factor)

*bias term*.

### C.2.2 Estimation of the asymptotic variance

Let us show now that, under the assumptions we made in Section C.2.1 and additional assumptions of similar nature, \(P_{n} D^{*} (\Phat_n)^2\) estimates consistently the asymptotic variance \(P_{0} D^{*} (P_{0})^2\). The proof hinges again on a decomposition of the difference between the two quantities as a sum of three terms: \[\begin{align} P_{n} D^{*} (\Phat_n)^2 - P_{0} D^{*} (P_{0})^2= & (P_{n} - P_{0}) \left(D^{*} (\Phat_n)^2 - D^{*} (P_{0})^2\right)\\ & + (P_{n} - P_{0}) D^{*} (P_{0})^2 + P_{0} (D^{*} \left(\Phat_n)^2 - D^{*} (P_{0})^2\right). \end{align}\]

We study the three terms in turn. Recall that \(X_n = o_{P_0}(1/\sqrt{n})\) means that \(P_0(\sqrt{n}|X_n| > t)\) converges to zero for all \(t>0\) as \(n\) goes to infinity.

In light of the study of the third term in Section C.2.1, if \(\|D^*(\Phat_n)^2 - D^*(P_0)^2\|_{P_{0}} = o_{P_0} (1)\) and if \(D^*(\Phat_n)^{2}\) falls (with probability tending to one) into a Donsker class, then the first term is \(o_{P_0}(1/\sqrt{n})\). Furthermore, if \(D^*(\Phat_n)\) falls (with probability tending to one) into a Donsker class, an assumption we made earlier, then so does \(D^*(\Phat_n)^{2}\). In addition, if \(\|D^*(\Phat_n) - D^*(P_0)\|_{P_{0}} = o_{P_0} (1)\), another assumption we made earlier, and if there exists a constant \(c>0\) such that \[\begin{equation} \sup_{n \geq 1} \|D^*(\Phat_n) + D^*(P_0)\|_{\infty} \leq c \tag{C.1} \end{equation}\] \(P_{0}\)-almost surely, then \(\|D^*(\Phat_n)^2 - D^*(P_0)^2\|_{P_{0}} = o_{P_0} (1)\) too because \[\begin{equation} \|D^*(\Phat_n)^2 - D^*(P_0)^2\|_{P_{0}} \leq c \|D^*(\Phat_n) - D^*(P_0)\|_{P_{0}}. \end{equation}\] The existence of such a constant \(c\) is granted whenever \(\ell\Gbar_{0}\) and \(\ell\Gbar_{n}\) are bounded away from zero. Note that the condition on \(\ell\Gbar_{n}\) can be inforced by us through the specification of algorithm \(\Algo_{\Gbar}\).

By the central limit theorem, \(\sqrt{n}\) times the second term converges in law to the centered Gaussian law with variance \(\Var_{P_{0}} (D^{*} (P_{0})(O)^2)\), which is finite whenever \(\ell\Gbar_{0}\) is bounded away from zero. By Theorem 2.4 in (Vaart 1998), the second term is thus \(O_{P_0} (1/\sqrt{n})\) hence \(o_{P_0} (1)\).

Finally, under assumption (C.1), the absolute value of the third term is smaller than \[\begin{equation}c P_{0} |D^*(\Phat_n) - D^*(P_0)| \leq c \|D^*(\Phat_n) - D^*(P_0)\|_{P_{0}} = o_{P_0}(1),\end{equation}\] where the inequality follows from the Cauchy-Schwarz inequality.

In conclusion, \(P_{n} D^{*} (\Phat_n)^2 - P_{0} D^{*} (P_{0})^2 = o_{P_0}(1)\), hence the result.

## C.3 Asymptotic negligibility of the remainder term

Recall that \(\|f\|_{P}^{2} \defq \Exp_{P} \left( f(O)^{2} \right)\) is the \(L_2(P)\)-norm of \(f\), a measurable function from \(\calO\) to \(\bbR\). Assume that for \(a= 0,1\), \(\ell\Gbar_{n}(a,W) \geq \delta > 0\) \(Q_{0,W}\)-almost everywhere.

The Cauchy-Schwarz inequality then implies that, for \(a = 0,1\),
\[\begin{equation*}\Rem_{P_0}(\Phat_n) \le \frac{2}{\delta} \max_{a=0,1}
\left( \|\Qbar_n (a,\cdot) - \Qbar_0 (a,\cdot)\|_{P_0} \right) \times
\|\Gbar_n - \Gbar_0\|_{P_0}.\end{equation*}\] Therefore, if for \(a=0,1\),
\[\begin{equation*}\|\Qbar_n(a,\cdot) - \Qbar_0(a,\cdot)\|_{P_0} =
o_{P_0}(n^{-1/4})\end{equation*}\] *and* \[\begin{equation*}\|\Gbar_n -
\Gbar_0\|_{P_0} = o_{P_0}(n^{-1/4}),\end{equation*}\] then
\[\begin{equation*}\Rem_{P_0}(\Phat_n) = o_{P_0}(n^{-1/2}).\end{equation*}\]

## C.4 Analysis of targeted estimators

### C.4.1 A basic fact on the influence curve equation

Recall the definition of \(D_{1}^{*}\) (3.4). For *any* estimator
\(\Qbar_n^*\) of \(\Qbar_0\) and a law \(P_n^{*}\) that is compatible with
\(\Qbar_n^*\) and \(Q_{n,W}\), it holds that
\[\begin{align*}
P_n D_1^*(P_n^*)
&= \frac{1}{n} \sum_{i=1}^n D_1^{*}(P_n^*)(O_i)\\
&=\frac{1}{n} \sum_{i=1}^n \left( \Qbar_n(1,W_i) - \Qbar_n(0,W_i) - \int
\left(\Qbar_n(1,w) - \Qbar_n(0,w)\right) dQ_{n,W}(w) \right) \\
&= \frac{1}{n}\sum_{i=1}^n \left( \Qbar_n(1,W_i) - \Qbar_n(0,W_i)\right) - \frac{1}{n}\sum_{i=1}^n
\left(\Qbar_n(1,W_i) - \Qbar_n(0,W_i)\right) = 0.
\end{align*}\]

### C.4.2 Fluctuation of the regression function along the fluctuation of a law

Let us resume the discussion where we left it at the end of Section 3.3.1. Let \(\Qbar\) be the conditional mean of \(Y\) given \((A,W)\) under \(P\). Set arbitrarily \(h \in H \setminus \{0\}\) and a measurable function \((w,a) \mapsto f(a,w)\) taking non-negative values. Applying repeatedly the tower rule yields the following equalities: \[\begin{align*} \Exp_{P_{h}} \left(f(A,W) Y\right) &= \Exp_{P} \left(f(A,W) Y (1 + h s(O))\right) \\ &= \Exp_{P} \left(f(A,W) \Exp_{P}\left(Y (1 + h s(O)) \middle|A,W\right)\right) \\ &= \Exp_{P} \left(f(A,W) \left(\Qbar(A,W) + h \Exp_{P}(Ys(O) | A,W) \right)\right) \\ &= \Exp_{P} \left(f(A,W) \frac{\Qbar(A,W) + h \Exp_{P}(Ys(O) | A,W)}{1 + h \Exp_{P}(s(O)|A,W)} \times \left(1 + h \Exp_{P}(s(O)|A,W)\right)\right).\end{align*}\] Now, (3.1) implies that the density of \((A,W)\) under \(P_{h}\) equals \(\left(1 + h \Exp_{P}(s(O)|A,W)\right)\) when it is evaluated at \((A,W)\). Therefore, the last inequality rewrites as \[\begin{equation*} \Exp_{P_{h}} \left(f(A,W) Y\right) = \Exp_{P_{h}} \left(f(A,W) \frac{\Qbar(A,W) + h \Exp_{P}(Ys(O) | A,W)}{1 + h \Exp_{P}(s(O)|A,W)}\right). \end{equation*}\] Since this equality is valid for an arbitary \((w,a) \mapsto f(a,w)\) with non-negative values, we can deduce from it that the conditional mean of \(Y\) given \((A,W)\) under \(P_h\) equals \[\begin{equation*}\frac{\Qbar(A,W) + h \Exp_{P}(Ys(O) | A,W)}{1 + h \Exp_{P}(s(O)|A,W)}.\end{equation*}\]

### C.4.3 Computing the score of a fluctuation of the regression function

Let us resume the discussion where we left it at the beginning of Section 10.2.2. Set \(\alpha, \beta \in \bbR\). The derivative of \(h \mapsto \expit(\alpha + \beta h)\) evaluated at \(h=0\) satisfies \[\begin{equation*}\frac{d}{dh} \left.\expit(\alpha + \beta h)\right|_{h=0} = \beta \expit(\alpha) (1 - \expit(\alpha)).\end{equation*}\] Therefore, for any \((w,a) \in [0,1] \times \{0,1\}\), \[\begin{align*}\frac{d}{dh} \left.\Qbar_{h} (a, w)\right|_{h=0} &= \frac{2a - 1}{\ell\Gbar(a,w)} \expit\left(\logit\left(\Qbar(a,w)\right)\right) \left[1 - \expit\left(\logit\left(\Qbar(a,w)\right)\right)\right]\\&= \frac{2a - 1}{\ell\Gbar(a,w)} \Qbar(a,w) \left(1 - \Qbar(a,w)\right).\end{align*}\] This justifies the last but one equality in (10.8).

Furthermore the same derivations that led to (10.8) also imply,
*mutatis mutandis*, that

\[\begin{equation*} \frac{d}{d h} \left. L_{y}(\Qbar_{h})(O)\right|_{h=0} = \frac{2A - 1}{\ell\Gbar(A, W)} \left(Y - \Qbar(A, W)\right). \tag{C.2}\end{equation*}\]

In this light, and in view of (3.2), we can think of \(\calQ(\Qbar, \Gbar)\) as a fluctuation of \(\Qbar\) in the direction of \[\begin{equation*}(w,a,y) \mapsto \frac{2a-1}{\ell\Gbar(a,w)} (y - \Qbar(a,w)).\end{equation*}\] Thus if \(P \in \calM\) is such that \(\Exp_{P}(Y|A,W) = \Qbar(A,W)\) and \(P(A=1|W) = \Gbar(W)\), then we can also think of \(\calQ(\Qbar, \Gbar)\) as a fluctuation of \(\Qbar\) in the direction of the second component \(D_{2}^{*}(P)\) of the efficient influence curve \(D^{*}(P)\) of \(\Psi\) at \(P\) (3.4).