SlowGuess/NeurIPS_2023_OCR huggingface dataset:
simulations for each  $n \in \{100, 150, 200, 250, 300, 350\}$ . We exhibit an illustration of the synthetic data from one of the simulations carried in Figure 5.
Real data. We use data obtained from the Infant Health and Development Program (IHDP) and compiled by Hill (2011), in which the covariates come from a randomized experiment studying the effects of specialist home visits on cognitive test scores. The propensity scores are unknown and hence they must be estimated. We work with the following 18 variables of the covariate set: ['bw', 'momage', 'nnhealth', 'birth.o', 'parity', 'moreprem', 'cigs', 'alcohol', 'ppvvt.IMP', 'bwg', 'female', 'mlt.birt', 'b.marry', 'livwho', 'language', 'whenpren', 'drugs', 'othstudy'], where the first nine are continuous and the last nine are discrete; we refer to Hill (2011) for a detailed presentation of the data set. We eliminate the rows in which there is a NaN for any of the variables considered, finishing with 908 observations, out of which 347 were treated. Further, we standardize the data such that every continuous variable has mean 0 and variance 1. We illustrate the data obtained after preprocessing in Figure 6.
We consider five scenarios with the IHDP data. For Scenarios I, II, III and IV, we generate the response variables similarly to the previous experiments:
$$
Y _ {0} ^ {*} = \beta^ {T} X + \epsilon_ {0}, Y _ {1} ^ {*} = \beta^ {T} X + b + \epsilon_ {1}, \tag {5}
$$
where  $\epsilon_0, \epsilon_1 \sim N(0, 0.5)$  are independent noises, and  $\beta = [1, \dots, 1]^T$ . We set  $b = 0$  and  $b = 1$  for Scenario I and Scenario II respectively. For scenario III, we set  $b = 2(2Z - 1)$ , where  $Z$  is an independent Bernoulli random variable  $Z \sim \text{Bernoulli}(0.5)$ . In scenario IV,  $b \sim \text{Uniform}(-4, 4)$ . In Scenario V, we take the IQ test (Stanford Binet) score, variable 'iqsb.36' of the data set, measured at the end of the intervention (age 3) as our response variable. For Scenario VI, we calculate the average treatment effect in Scenario V using Causal Forests (obtaining 0.0035 i.e. a positive shift) and subtract it to the IQ test score of those who are treated, thus obtaining a distribution of the IQ test scores with zero average treatment effect. We illustrate the data of Scenarios V and VI in Figure 6. For each of the scenarios, we consider the tests AIPW-xKTE, IPW-xKTE, Causal Forests, BART and Baseline-AIPW on 500 bootstrapped subsets to estimate the rejection rates. For AIPW-xKTE and IPW-xKTE, we estimate  $\hat{\pi}$  using a regularized logistic regression with the regularization term set to 1e-6. For AIPW-xKTE, the mean embedding regressions are fitted as conditional mean embeddings.
We sample split for estimating the conditional mean embeddings, but for the propensity scores, we use the whole training data. While we stated Theorem 4.1 imposing condition (iv) or condition (v) on  $\hat{\phi}^{(r)}$  (common practice in causal inference for ease of presentation, where one usually imposes
Figure 6: Illustration of the IHDP data for Scenarios V and VI. We display scatter plots of the outcome against the first principal component and histograms of the outcomes, both grouped by treatment.
Figure 7: Illustration of 500 simulations of the AIPW-xKTE with non-linear effects under the null: (A) Histogram of AIPW-xKTE alongside the pdf of a standard normal for  $n = 500$ , (B) Normal Q-Q plot of AIPW-xKTE for  $n = 500$ , (C) Empirical size of AIPW-xKTE against different sample sizes. The figures show the same Gaussian behaviour of the statistic under the null as in the linear setting.
sample splitting or a Donsker condition), Theorem 4.1 holds as long as sample splitting is conducted for those estimators  $\hat{\pi}^{(r)},\hat{\beta}_a^{(r)}$  which may overfit (analogous case in the standard doubly robust estimator). In our case, the propensity is modeled by L2 regularized logistic regression, which is simple and cannot overfit when the dimension is fixed and  $n\to \infty$
All the kernels considered on  $\mathcal{V}$  for AIPW-xKTE, IPW-xKTE, and KTE are RBF i.e.  $k_{y}(y_{1},y_{2}) =$ $\exp (\nu_y|y_1 - y_2|^2)$ , with scale parameter  $\nu_{y}$  chosen by the median heuristic. For AIPW-xKTE, the mean embedding regressions are fitted as conditional mean embeddings, and we conduct sample splitting in order to train them. The kernel considered for such conditional mean embeddings on  $\mathcal{X}$  is also RBF  $k_{x}(x_{1},x_{2}) = \exp (\nu_{x}|x_{1} - x_{2}|^{2})$  with scale parameter  $\nu_{x}$  chosen by the median heuristic as well. The regularization parameter  $\lambda$  was taken to be equal to  $\nu_{x}$ .
B.2 Further experiments
In this subsection, we investigate the performance of the tests in non-linear settings. We extend the simulations with synthetic data presented in Subsection B.1, defining the potential outcomes
$$
Y _ {0} ^ {*} = \cos (\beta^ {T} X) + \epsilon_ {0}, \quad Y _ {1} ^ {*} = \cos (\beta^ {T} X) + b + \epsilon_ {1}, \tag {6}
$$
where  $\beta, b, \epsilon_0$  and  $\epsilon_{1}$  are defined as described in Subsection B.1. The different scenarios and the remaining parameters are also defined as in Subsection B.1.
Figure 7 displays the behaviour of AIPW-xKTE in Scenario I (under the null). Analogously to the linear setting, we note that AIPW-xKTE presents a Gaussian behaviour under the null, which leads to a well-calibrated test. The performance of the tests in Scenarios II, III and IV is displayed in Figure 8. In this non-linear case, AIPW-xKTE proves even more competitive in Scenario II, while retaining power in Scenarios III and IV. In short, the test proves valid (as expected) in the non-linear case, and the comments from Section 5 equally apply.
Figure 8: True positive rates of 500 simulations of the tests in Scenarios II, III, and IV with non-linear effects. AIPW-xKTE shows notable true positive rates in every scenario, as in the linear setting.
C Proofs
We present in this appendix the proof of the main result of the paper, Theorem 4.1. For this purpose, we introduce the notation used in Subsection C.1 followed by the exposition of previously known results in Subsection C.2. We present a sequence of extensions of such results to the vector-valued scenario in Subsection C.3, which will be used to prove the main theorem of the paper in Subsection C.4.
We use standard big-oh and little-oh notation, i.e.,  $X_{n} = O_{\mathbb{P}}(r_{n})$  means  $X_{n} / r_{n}$  is bounded in probability and  $X_{n} = o_{\mathbb{P}}(r_{n})$  means  $X_{n} / r_{n} \stackrel{p}{\to} 0$ . We use  $\mathbb{P}_n$  for sample averages i.e.  $\mathbb{P}_n(f) = \mathbb{P}_n\{f(Z)\} = \frac{1}{n}\sum_if(Z_i)$ . For a potentially random function  $\widehat{f}$ , we denote  $\mathbb{P}(\widehat{f}) = \mathbb{P}\{\widehat{f}(Z)\} = \int \widehat{f}(z)d\mathbb{P}(z)$ , and  $\| \widehat{f} \|^{2} = \int \widehat{f}(z)^{2}d\mathbb{P}(z)$  represents the squared  $L_2(\mathbb{P})$  norm.
Given a Hilbert space  $\mathcal{H}$  and a  $\mathcal{H}$ -valued  $\omega(z) \in \mathcal{H}$ , we denote its norm in the Hilbert space by  $\| \omega(z) \|_{\mathcal{H}}$ . We let  $\mathbb{P}_n(\omega) = \mathbb{P}_n\{\omega(Z)\} = \frac{1}{n} \sum_i \omega(Z_i)$  and  $\mathbb{P}(\widehat{\omega}) = \mathbb{P}\{\widehat{\omega}(Z)\} = \int \widehat{\omega}(z) d\mathbb{P}(z)$  (Bochner integral). Furthermore,  $\| \widehat{\omega} \|^2 = \int \| \widehat{\omega}(z) \|_{\mathcal{H}}^2 d\mathbb{P}(z)$  denotes the squared  $L_2(\mathbb{P})$  norm of the  $\mathcal{H}$ -valued function norm, and  $\| \cdot \|_{\mathrm{HS}}$  the Hilbert-Schmidt norm of an operator.
C.2 Previously known results
We start by introducing a generalization of the formula that relates the mean, variance and second moment of a real-valued random variable to  $\mathcal{H}$ -valued random variables.
Lemma C.1 (Hsing and Eubank (2015, Theorem 7.2.2)). Let  $\omega \in \mathcal{H}$  be a random variable such that  $\mathbb{E}||\omega ||_{\mathcal{H}}^2 < \infty$ . Then
$$
\mathbb {E} \| \omega - \mathbb {E} [ \omega ] \| _ {\mathcal {H}} ^ {2} = \mathbb {E} \| \omega \| _ {\mathcal {H}} ^ {2} - \| \mathbb {E} [ \omega ] \| _ {\mathcal {H}} ^ {2}.
$$
Furthermore, one may retrieve the expectation of the product of projections of the  $\mathcal{H}$ -valued random variables through the covariance operator, as exhibited in the following lemma.
Lemma C.2. Let  $\omega \in \mathcal{H}$  be a random variable such that  $\mathbb{E}[\omega] = 0$  and  $\mathbb{E}\|\omega\|_{\mathcal{H}}^2 < \infty$ . Then
$$
\mathbb {E} [ \langle \omega , f \rangle \langle \omega , g \rangle ] = \langle C f, g \rangle , \quad f, g \in \mathcal {H},
$$
where  $C = \mathbb{E}[\omega \otimes \omega]$  is the covariance operator.
Next, we present the generalization of the central limit theorem to  $\mathcal{H}$ -valued random variables.
Theorem C.3 (Bosq (2000, Theorem 2.7)). Let  $(\omega_{i}, i \geq 1)$  be a sequence of iid random variables in a separable Hilbert space  $\mathcal{H}$ . If  $m = \mathbb{E}[\omega_1]$  and  $\mathbb{E}[\| \omega_1 \|^2] < \infty$ , then
$$
\frac {1}{\sqrt {n}} \sum_ {i = 1} ^ {n} (\omega_ {i} - m) \stackrel {d} {\rightarrow} \xi ,
$$
where  $\xi$  is a Gaussian random element with mean zero and covariance operator  $C = \mathbb{E}[\omega_1\otimes \omega_1]$  i.e.
$$
\xi \sim N (0, \mathbb {E} [ \omega_ {1} \otimes \omega_ {1} ]).
$$
Moreover, Gaussian  $\mathcal{H}$ -valued random variables admit the following expansion in the orthonormal system defined by its covariance operator.
Lemma C.4 (Horváth and Kokoszka (2012, Equation 2.8)). A normally distributed function  $\xi$  in a separable Hilbert space with a covariance operator  $C$  admits the expansion
$$
\xi \stackrel {d} {=} \sum_ {i} ^ {\infty} \sqrt {\lambda_ {j}} N _ {j} v _ {j},
$$
where  $N_{j}$  are independent standard normal distributions,  $(\lambda_j,j\geq 1)$  is the sequence of eigenvalues of the covariance operator  $C$ , and  $(v_{j},j\geq 1)$  is a sequence of orthonormal eigenfunctions of the covariance operator  $C$ .
We now present a result concerning the asymptotic behaviour of an empirical process given that the considered random functions belong to a Donsker class. For a full discussion on Donsker classes and empirical processes, we refer the reader to Chapter 18 and Chapter 19 of Van der Vaart (2000).
Lemma C.5 (Van der Vaart (2000, Lemma 19.24)). Suppose that  $\mathcal{F}$  is a  $\mathbb{P}$ -Donsker class of measurable (real-valued) functions and  $\hat{f}_n$  is a sequence of random functions that take their values in  $\mathcal{F}$  such that  $\| \hat{f}_n - f\|^2 \xrightarrow{p} 0$  for some  $f \in L_2(\mathbb{P})$ . Then
$$
\left(\mathbb {P} _ {n} - \mathbb {P}\right) \left(\widehat {f} _ {n} - f\right) = o _ {\mathbb {P}} \left(\frac {1}{\sqrt {n}}\right).
$$
However, the concept of Donsker class is fairly restrictive. In fact, the intuition behind the nice behaviour exhibited by Donsker classes in empirical process is their inability to overfit, due to their constricted flexibility. For instance, commonly used estimators such as random forests do not belong to a Donsker class. In order to circumvent this problem, sample splitting proves very useful. The following lemma exhibits the power of sample splitting regardless of the flexibility of the estimators.
Lemma C.6 (Kennedy et al. (2020, Lemma 2)). Let  $\widehat{f}(z)$  be a function estimated from a sample  $Z^{N} = (Z_{n+1},\ldots,Z_{N})$ , and let  $\mathbb{P}_n$  denote the empirical measure over  $(Z_{1},\ldots,Z_{n})$ , which is independent of  $Z^{N}$ . Then
$$
(\mathbb {P} _ {n} - \mathbb {P}) (\widehat {f} - f) = O _ {\mathbb {P}} \left(\frac {\| \widehat {f} - f \|}{\sqrt {n}}\right).
$$
Proof. First note that, conditional on  $Z^{N}$ , the term in question has mean zero since
$$
\begin{array}{l} \mathbb {E} \left\{\left(\mathbb {P} _ {n} - \mathbb {P}\right) (\widehat {f} - f) \mid Z ^ {N} \right\} = \mathbb {E} \left\{\mathbb {P} _ {n} (\widehat {f} - f) \mid Z ^ {N} \right\} - \mathbb {P} (\widehat {f} - f) \\ = \mathbb {E} (\widehat {f} - f \mid Z ^ {N}) - \mathbb {P} (\widehat {f} - f) \\ = \mathbb {P} (\hat {f} - f) - \mathbb {P} (\hat {f} - f) \\ = 0. \\ \end{array}
$$
The conditional variance is
$$
\operatorname {v a r} \left\{\left(\mathbb {P} _ {n} - \mathbb {P}\right) (\widehat {f} - f) \mid Z ^ {N} \right\} = \operatorname {v a r} \left\{\mathbb {P} _ {n} (\widehat {f} - f) \mid Z ^ {N} \right\} = \frac {1}{n} \operatorname {v a r} (\widehat {f} - f \mid Z ^ {N}) \leq \| \widehat {f} - f \| ^ {2} / n.
$$
Therefore by iterated expectation and Chebyshev's inequality we have
$$
\mathbb {P} \left\{\frac {| (\mathbb {P} _ {n} - \mathbb {P}) (\widehat {f} - f) |}{\| \widehat {f} - f \| / \sqrt {n}} \geq t \right\} = \mathbb {P} \left[ \mathbb {E} \left\{\frac {| (\mathbb {P} _ {n} - \mathbb {P}) (\widehat {f} - f) |}{\| \widehat {f} - f \| / \sqrt {n}} \geq t \Big | Z ^ {N} \right\} \right] \leq \frac {1}{t ^ {2}}.
$$
Thus for any  $\epsilon > 0$  we can pick  $t = 1 / \sqrt{\epsilon}$  so that the probability above is no more than  $\epsilon$ , which yields the result.
These two lemmas allow for proving Theorem 3.1. Recall that this theorem exhibits the empirical mean-like behaviour of doubly robust estimators. This will be key when proving the main result of the paper.
Proof of Theorem 3.1. We have that  $\hat{\psi}_{\mathrm{DR}} - \psi = Z^{*} + T_{1} + T_{2}$ , where
$$
Z ^ {*} = (\mathbb {P} _ {n} - \mathbb {P}) f, \quad T _ {1} = (\mathbb {P} _ {n} - \mathbb {P}) (\hat {f} - f), \quad T _ {2} = \mathbb {P} (\hat {f} - f).
$$
The consistency of  $f$  in  $L_{2}$  norm, together with either the Donsker condition (Lemma C.5) or sample splitting (Lemma C.6), ensures that  $T_{1} = o_{\mathbb{P}}(1 / \sqrt{n})$ . For  $T_{2} = \mathbb{P}(\hat{f} - f)$  and  $f = f_{1} - f_{0}$ , we have
$$
\begin{array}{l} \left| \mathbb {P} \left(\hat {f} _ {1} - f _ {1}\right) \right| = \left| \mathbb {P} \left\{\frac {A}{\hat {\pi}} \left(Y - \hat {\theta} _ {A}\right) + \hat {\theta} _ {1} - \theta_ {1} \right\} \right| = \left| \mathbb {P} \left\{\left(\frac {\pi}{\hat {\pi}} - 1\right) \left(\theta_ {1} - \hat {\theta} _ {1}\right) \right\} \right| \\ \leq \frac {1}{\epsilon} \mathbb {P} \left\{\left| \pi - \hat {\pi} \right| \left| \theta_ {1} - \hat {\theta} _ {1} \right| \right\} \leq \frac {1}{\epsilon} \| \pi - \hat {\pi} \| \| \theta_ {1} - \hat {\theta} _ {1} \|. \\ \end{array}
$$
Same logic applies for  $|\mathbb{P}(\hat{f}_0 - f_0)| \leq \frac{1}{1 - \epsilon} \| \pi - \hat{\pi} \| \| \theta_0 - \hat{\theta}_0 \|$ . Therefore  $T_2 = o_{\mathbb{P}}(1 / \sqrt{n})$  and the result follows.
We have exhibited the proofs of Lemma C.6 and Theorem 3.1 for sake of completeness. We will extend and prove the analogous results in the  $\mathcal{H}$ -valued setting in Subsection C.3, which will be needed in demonstrating the main result of the paper. We refer the reader to Kennedy (2022) for a full discussion and exhibition of the aforementioned causal inference results. Lastly, we exhibit the definition of a Glivenko-Cantelli class, which is used in Theorem 4.1.
Definition C.7 (Glivenko-Cantelli). We say that a class of integrable real-valued functions  $\mathcal{F}$  is a Glivenko-Cantelli class for  $\mathbb{P}$  if
$$
\sup  _ {f \in \mathcal {F}} \left| \frac {1}{n} \sum_ {i = 1} ^ {n} f (X _ {i}) - \mathbb {E} [ f (X) ] \right|
$$
converges to zero in probability as  $n\to \infty$
C.3 Extension of the results to (infinite-dimensional) vector-valued outcomes
We introduce the extension of Theorem 3.1 to the  $\mathcal{H}$ -valued outcome setting. For such a goal, we also generalize Lemma C.6 to the  $\mathcal{H}$ -valued scenario and we comment on asymptotically equicontinuous empirical processes Park and Muandet (2023), which will substitute the Donsker class condition in the generalized version of Theorem 3.1.
We start by presenting the generalization of Lemma C.6 to the functional setting.
Lemma C.8 (Extension of Lemma C.6 to  $\mathcal{H}$ -valued outcomes). Let  $\mathcal{H}$  be a Hilbert space. Let  $\widehat{\omega}(z)$  be a function estimated from a sample  $Z^{N} = (Z_{n+1},\dots,Z_{N})$  such that  $\widehat{\omega}(z) \in \mathcal{H}$  for all  $z$ , and let  $\mathbb{P}_n$  denote the empirical measure over  $(Z_1,\dots,Z_n)$ , which is independent of  $Z^{N}$ . Then
$$
\| (\mathbb {P} _ {n} - \mathbb {P}) (\widehat {\omega} - \omega) \| _ {\mathcal {H}} = O _ {\mathbb {P}} \left(\frac {\| \widehat {\omega} - \omega \|}{\sqrt {n}}\right).
$$
$$
\begin{array}{l} \mathbb {E} \left\{\left(\mathbb {P} _ {n} - \mathbb {P}\right) (\widehat {\omega} - \omega) \mid Z ^ {N} \right\} = \mathbb {E} \left\{\mathbb {P} _ {n} (\widehat {\omega} - \omega) \mid Z ^ {N} \right\} - \mathbb {P} (\widehat {\omega} - \omega) \\ = \mathbb {E} (\widehat {\omega} - \omega \mid Z ^ {N}) - \mathbb {P} (\widehat {\omega} - \omega) \\ = \mathbb {P} (\widehat {\omega} - \omega) - \mathbb {P} (\widehat {\omega} - \omega) \\ = 0. \\ \end{array}
$$
The conditional expectation of the squared norm is
$$
\begin{array}{l} \mathbb {E} \Big \{\| (\mathbb {P} _ {n} - \mathbb {P}) (\widehat {\omega} - \omega) \| _ {\mathcal {H}} ^ {2} \Big | Z ^ {N} \Big \} = \mathbb {E} \Big \{\langle (\mathbb {P} _ {n} - \mathbb {P}) (\widehat {\omega} - \omega), (\mathbb {P} _ {n} - \mathbb {P}) (\widehat {\omega} - \omega) \rangle_ {\mathcal {H}} \Big | Z ^ {N} \Big \} \\ = \mathbb {E} \Big \{\langle \frac {1}{n} \sum_ {i = 1} ^ {n} [ (\widehat {\omega} - \omega) (Z _ {i}) - \mathbb {P} (\widehat {\omega} - \omega) ], \frac {1}{n} \sum_ {j = 1} ^ {n} [ (\widehat {\omega} - \omega) (Z _ {j}) - \mathbb {P} (\widehat {\omega} - \omega) ] \rangle_ {\mathcal {H}} \Big | Z ^ {N} \Big \} \\ = \frac {1}{n ^ {2}} \sum_ {i, j = 1} ^ {n} \mathbb {E} \Big \{\langle (\widehat {\omega} - \omega) (Z _ {i}) - \mathbb {P} (\widehat {\omega} - \omega), (\widehat {\omega} - \omega) (Z _ {j}) - \mathbb {P} (\widehat {\omega} - \omega) \rangle_ {\mathcal {H}} \Big | Z ^ {N} \Big \} \\ \stackrel {(i)} {=} \frac {1}{n ^ {2}} \sum_ {i = 1} ^ {n} \mathbb {E} \left\{\langle (\widehat {\omega} - \omega) (Z _ {i}) - \mathbb {P} (\widehat {\omega} - \omega), (\widehat {\omega} - \omega) (Z _ {i}) - \mathbb {P} (\widehat {\omega} - \omega) \rangle_ {\mathcal {H}} \mid Z ^ {N} \right\} \\ = \frac {1}{n ^ {2}} \sum_ {i = 1} ^ {n} \mathbb {E} \left\{\| (\widehat {\omega} - \omega) (Z _ {i}) - \mathbb {P} (\widehat {\omega} - \omega) \| _ {\mathcal {H}} ^ {2} \mid Z ^ {N} \right\} \\ = \frac {1}{n} \mathbb {E} \left\{\| (\widehat {\omega} - \omega) (Z _ {1}) - \mathbb {P} (\widehat {\omega} - \omega) \| _ {\mathcal {H}} ^ {2} \Big | Z ^ {N} \right\} \\ \stackrel {(i i)} {=} \frac {1}{n} \left[ \mathbb {E} \left\{\| (\widehat {\omega} - \omega) (Z _ {1}) \| _ {\mathcal {H}} ^ {2} \mid Z ^ {N} \right\} - \left\{\| \mathbb {P} (\widehat {\omega} - \omega) \| _ {\mathcal {H}} ^ {2} \mid Z ^ {N} \right\} \right] \\ = \frac {1}{n} \| \widehat {\omega} - \omega \| ^ {2} - \frac {1}{n} \left\{\| \mathbb {P} (\widehat {\omega} - \omega) \| _ {\mathcal {H}} ^ {2} \Big | Z ^ {N} \right\} \\ \leq \frac {1}{n} \| \widehat {\omega} - \omega \| ^ {2}, \\ \end{array}
$$
where (i) is obtained given independence of  $Z_{i}, Z_{j}$  when  $i \neq j$  and  $\mathbb{E}\left\{(\mathbb{P}_n - \mathbb{P})(\widehat{\omega} - \omega) \Big| Z^N\right\} = 0$ , and (ii) by Lemma C.1. Therefore by iterated expectation and Markov's inequality we have
$$
\begin{array}{l} \mathbb {P} \left\{\frac {\| (\mathbb {P} _ {n} - \mathbb {P}) (\widehat {\omega} - \omega) \| _ {\mathcal {H}}}{\| \widehat {\omega} - \omega \| / \sqrt {n}} \geq t \right\} = \mathbb {P} \left[ \mathbb {E} \left\{\frac {\| (\mathbb {P} _ {n} - \mathbb {P}) (\widehat {\omega} - \omega) \| _ {\mathcal {H}}}{\| \widehat {\omega} - \omega \| / \sqrt {n}} \geq t \middle | Z ^ {N} \right\} \right] \\ = \mathbb {P} \left[ \mathbb {E} \left\{\frac {\| (\mathbb {P} _ {n} - \mathbb {P}) (\widehat {\omega} - \omega) \| _ {\mathcal {H}} ^ {2}}{\| \widehat {\omega} - \omega \| ^ {2} / n} \geq t ^ {2} \Big | Z ^ {N} \right\} \right] \\ \leq \frac {1}{t ^ {2}}. \\ \end{array}
$$
Note that the former result implies that, if  $\| \widehat{\omega} - \omega \| \stackrel{p}{\to} 0$ , then
$$
\| (\mathbb {P} _ {n} - \mathbb {P}) (\widehat {\omega} - \omega) \| _ {\mathcal {H}} = o _ {\mathbb {P}} \left(\frac {1}{\sqrt {n}}\right),
$$
given that we conduct sample splitting. However, sample splitting might not be necessary if our estimators are not flexible enough. Splitting the data might result in a loss in power, hence we are interested in stating another sufficient condition that will lead the empirical term  $(\mathbb{P}_n - \mathbb{P})(\hat{\omega} - \omega)$  to be asymptotically negligible without conducting sample splitting. A direct extension of the Donsker class condition stated in Theorem 3.1 to  $\mathcal{H}$ -valued outcomes is not possible, given that bounding entropies usually make explicit use of the fact that  $\mathbb{R}$  is totally ordered (Park and Muandet, 2023). However, Park and Muandet (2023) defines asymptotic equicontinuity, which is precisely the condition which we will require in the extension of Theorem 3.1 to the functional setting.
Definition C.9 (Asymptotic equicontinuity). We say that the empirical process  $\{\nu_n(\omega) = \sqrt{n} (\mathbb{P}_n - \mathbb{P})\omega : \omega \in \mathcal{G}\}$  with values in  $\mathcal{H}$  and indexed by  $\mathcal{G}$  is asymptotic equicontinuous at  $\omega_0 \in \mathcal{G}$  if, for every sequence  $\{\hat{\omega}_n\} \subset \mathcal{G}$  with  $\| \hat{\omega}_n - \omega_0 \| \xrightarrow{p} 0$ , we have
$$
\left\| \nu_ {n} \left(\hat {\omega} _ {n}\right) - \nu_ {n} \left(\omega_ {0}\right) \right\| _ {\mathcal {H}} \xrightarrow {p} 0. \tag {7}
$$
Note that (7) is equivalent to  $(\mathbb{P}_n - \mathbb{P})(\hat{\omega} - \omega) = o_{\mathbb{P}}\left(\frac{1}{\sqrt{n}}\right)$ . Park and Muandet (2023) gives sufficient conditions for asymptotic equicontinuity to hold, as well as stating examples of classes that attain asymptotic equicontinuity. We are now ready to present the extension of Theorem 3.1 to the  $\mathcal{H}$ -valued setting.
Theorem C.10 (Extension of Theorem 3.1 to  $\mathcal{H}$ -valued outcomes). Let  $\phi(z) = \left\{\frac{a}{\pi(x)} - \frac{1 - a}{1 - \pi(x)}\right\} \{y - \beta_a(x)\} + \beta_1(x) - \beta_0(x)$ , where  $y$  and  $\beta$  are  $\mathcal{H}$ -valued, so that  $\Psi = \mathbb{E}\{\phi(Z)\}$  is the average treatment effect. Suppose that
-  $\hat{\phi}$  is constructed from an independent sample or the respective empirical process is asymptotically equicontinuous at  $\phi$ .
-  $||\hat{\phi} - \phi|| = o_{\mathbb{P}}(1)$ .
Also assume that  $\mathbb{P}(\hat{\pi}\in [\epsilon ,1 - \epsilon ]) = 1$ . Then if  $||\hat{\pi} -\pi ||\sum_{a}||\hat{\beta}_{a} - \beta_{a}|| = o_{\mathbb{P}}\big(\frac{1}{\sqrt{n}}\big)$  it follows that
$$
\hat {\Psi} _ {D R} - \Psi = (\mathbb {P} _ {n} - \mathbb {P}) \phi (Z) + \xi_ {n},
$$
where  $\| \xi_n\|_{\mathcal{H}} = o_{\mathbb{P}}\bigl (\frac{1}{\sqrt{n}}\bigr)$ . Hence, it is root-n consistent.
Proof. We have that  $\hat{\Psi}_{\mathrm{DR}} - \Psi = Z^{*} + T_{1} + T_{2}$ , where
$$
Z ^ {*} = (\mathbb {P} _ {n} - \mathbb {P}) \phi , \quad T _ {1} = (\mathbb {P} _ {n} - \mathbb {P}) (\hat {\phi} - \phi), \quad T _ {2} = \mathbb {P} (\hat {\phi} - \phi).
$$
The consistency of  $\phi$  in  $L_{2}$  norm, together with either asymptotic equicontinuity (Definition C.9) or sample splitting (Lemma C.8), ensures that  $\| T_1\|_{\mathcal{H}} = o_{\mathbb{P}}(1 / \sqrt{n})$ . For  $T_{2} = \mathbb{P}(\hat{\phi} -\phi)$  and
$\phi = \phi_{1} - \phi_{0}$  , we have
$$
\begin{array}{l} \| \mathbb {P} (\hat {\phi} _ {1} - \phi_ {1}) \| _ {\mathcal {H}} = \| \mathbb {P} \{\frac {A}{\hat {\pi}} (Y - \hat {\beta} _ {A}) + \hat {\beta} _ {1} - \beta_ {1} \} \| _ {\mathcal {H}} \leq \mathbb {P} \{\| (\frac {\pi}{\hat {\pi}} - 1) (\beta_ {1} - \hat {\beta} _ {1}) \| _ {\mathcal {H}} \} \\ \leq \frac {1}{\epsilon} \mathbb {P} \left\{\left| \pi - \hat {\pi} \right| \left\| \beta_ {1} - \hat {\beta} _ {1} \right\| _ {\mathcal {H}} \right\} \leq \frac {1}{\epsilon} \| \pi - \hat {\pi} \| \| \beta_ {1} - \hat {\beta} _ {1} \|. \\ \end{array}
$$