SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Diego Martinez-Taboada
Department of Statistics and Data Science
diegomar@andrew.cmu.edu
Machine Learning Department
aramdas@stat.cmu.edu
Edward H. Kennedy
edward@stat.cmu.edu
The average treatment effect, which is the difference in expectation of the counterfactuals, is probably the most popular target effect in causal inference with binary treatments. However, treatments may have effects beyond the mean, for instance decreasing or increasing the variance. We propose a new kernel-based test for distributional effects of the treatment. It is, to the best of our knowledge, the first kernel-based, doubly-robust test with provably valid type-I error. Furthermore, our proposed algorithm is computationally efficient, avoiding the use of permutations.
In the context of causal inference, potential outcomes (Rubin, 2005) are widely used to address counterfactual questions (e.g. what would have happened had some intervention been performed?). This framework considers
$$
(X, A, Y) \sim \mathbb {P},
$$
where  $X$  and  $Y$  represent the covariates and outcome respectively, and  $A \in \{0,1\}$  is a binary treatment. Furthermore, the following three conditions are assumed:
i) (Consistency)  $Y = AY_{1}^{*} + (1 - A)Y_{0}^{*}$  (where  $Y_{1}^{*}, Y_{0}^{*}$  are the potential outcomes).
ii) (No unmeasured confounding)  $Y_0^*, Y_1^* \perp A|X$ .
iii) (Overlap) For some  $\epsilon > 0$ , we have  $\epsilon < \pi(X) \coloneqq \mathbb{P}_{A|X}(A = 1|X) < 1 - \epsilon$  almost surely.
Such assumptions allow for identification of causal target parameters. For instance, one may be interested in the average variance-weighted treatment effects (Robins et al., 2008; Li et al., 2011), stochastic intervention effects (Muñoz and Van Der Laan, 2012; Kennedy, 2019), or treatment effect bounds (Richardson et al., 2014; Luedtke et al., 2015). However, most of the literature focuses on estimation and inference of the average treatment effect (Imbens, 2004; Hernan and Robins, 2020), defined as the difference in expectation of the potential outcomes
$$
\psi = \mathbb {E} [ Y _ {1} ^ {*} - Y _ {0} ^ {*} ].
$$
Given  $(X_{i},A_{i},Y_{i})_{i = 1}^{N}\sim (X,A,Y)$ , there are three widespread estimators of  $\psi$ . First, the plug-in (PI) estimator:
$$
\hat {\psi} _ {\mathrm {P I}} = \frac {1}{N} \sum_ {i = 1} ^ {N} \left\{\hat {\theta} _ {1} (X _ {i}) - \hat {\theta} _ {0} (X _ {i}) \right\},
$$
where  $\hat{\theta}_1(X),\hat{\theta}_0(X)$  estimate  $\mathbb{E}[Y_1^* |X],\mathbb{E}[Y_0^* |X]$ . Second, the inverse propensity weighting (IPW) estimator:
$$
\hat {\psi} _ {\mathrm {I P W}} = \frac {1}{N} \sum_ {i = 1} ^ {N} \left\{\frac {Y _ {i} A _ {i}}{\hat {\pi} (X _ {i})} - \frac {Y _ {i} (1 - A _ {i})}{1 - \hat {\pi} (X _ {i})} \right\},
$$
where  $\hat{\pi}(X)$  estimates the propensity scores  $\mathbb{P}[A = 1|X]$ . Third, the so-called Augmented Inverse Propensity Weighted (AIPW) estimator:
$$
\hat {\psi} _ {\mathrm {A I P W}} = \frac {1}{N} \sum_ {i = 1} ^ {N} \left\{\hat {\theta} _ {1} (X _ {i}) - \hat {\theta} _ {0} (X _ {i}) + \left(\frac {A _ {i}}{\hat {\pi} (X _ {i})} - \frac {1 - A _ {i}}{1 - \hat {\pi} (X _ {i})}\right) \left(Y _ {i} - \hat {\theta} _ {A _ {i}} (X _ {i})\right) \right\}.
$$
Under certain conditions (e.g., consistent nuisance estimation at  $n^{-1/4}$  rates), the asymptotic mean squared error of the AIPW estimator is smaller than that of the IPW and PI estimator, and minimax optimal in a local asymptotic sense (Kennedy, 2022), hence it has become increasingly popular in the last decade. The AIPW estimator is often referred to as the doubly-robust estimator. We highlight that double-robustness is an intriguing property of an estimator that makes use of two models, in which the estimator is consistent even if only one of the two models is well-specified and the other may be misspecified; we refer the reader to Kang and Schafer (2007) for a discussion on doubly-robust procedures.
However, the treatment might have effects beyond the mean, for instance in the variance or skewness of the potential outcome. The average treatment effect will prove insufficient in this case. Consequently, one may be interested in testing whether the treatment has any effect in the distribution of the outcome. This question naturally arises in a variety of applications. For instance, one may want to check whether there is any difference between a brand-name drug and its generic counterpart, or understand whether a treatment simply shifts the distribution of the outcome (or, in turn, it also affects higher order moments).
In this work, we revisit the problem of testing the null hypothesis  $H_0: P_{Y_1^*} = P_{Y_0^*}$  against  $H_1: P_{Y_1^*} \neq P_{Y_0^*}$ . We propose a distributional treatment effect test based on kernel mean embeddings and the asymptotic behaviour of the AIPW estimator. Our contributions are three-fold:
- Up to our knowledge, we propose the first kernel-based distributional test to allow for doubly-robust estimators with provably valid type-I error.
- The proposed distributional treatment effect test is permutation-free, which makes it computationally efficient.
- We empirically test the power and size of the proposed test, showing the substantial benefits of the doubly robust approach.
Distributional treatment effects have been addressed from a variety of points of view. Abadie (2002) was one of the first works to propose to test distributional hypothesis attempting to estimate the counterfactual cumulative distribution function (cdf) of the outcome of the treated and untreated. Chernozhukov et al. (2013) proposed to regress the cdf after splitting the outcome in a grid. Further contributions followed with alike cdf-based approaches (Landmesser, 2016; Diaz, 2017).
Other approaches to the problem include focusing on the probability density function (pdf) instead of the cdf. Robins and Rotnitzky (2001) introduced a doubly robust kernel estimator for the counterfactual density, while Westling and Carone (2020) proposed to conduct density estimation under a monotone density assumption. Kim et al. (2018) and Kennedy et al. (2021) suggested to compute  $L^p$  distances between the pdf of the outcome distribution for the different counterfactuals. Conditional distributional treatment effects have also been addressed, with Shen (2019) proposing to estimate the cdf of the counterfactuals for each value to be conditioned on.
On the other hand, kernel methods have recently gained more and more attention in the context of causal inference. Kernel-based two-stage instrumental variable regression was proposed in Singh et al. (2019), while Singh et al. (2020) presented estimators based on kernel ridge regression for nonparametric causal functions such as dose, heterogeneous, and incremental response curves. Furthermore, Singh et al. (2021) conducted mediation analysis and dynamic treatment effect using kernel-based regressors as nuisance functions. Causal inference with treatment measurement error i.e. when the cause is corrupted by error was addressed in Zhu et al. (2022) using kernel mean embeddings to learn the latent characteristic function.
However, kernel mean embeddings for distributional representation were not suggested in the causal inference literature until Muandet et al. (2021) proposed to use an IPW estimator to estimate the average treatment effect on the embedding, which leads to a kernel-based distributional treatment effect test based on this embedding and the MMD. With a very similar motivation, Park et al. (2021) proposed a test for conditional distributional treatment effects based on kernel conditional mean embeddings. In a preprint, Fawkes et al. (2022) extended the work to AIPW estimators, however no theoretical guarantees regarding type-1 error control of the proposed tests were provided.
Finally, the kernel-based tests used in the aforementioned cases involve test statistics that are degenerate U-statistics under the null, hence obtaining theoretical p-values of the statistic is not possible. In turn, Kim and Ramdas (2023) proposed the idea of cross U-statistics, which is based on splitting the data for achieving a normal asymptotic distribution after studentization. Similarly, Shekhar et al. (2022) exploited sample splitting for proposing a permutation-free kernel two sample test.
In this section, we introduce the concepts that the proposed test for distributional treatment effects is mainly based on: Maximum Mean Discrepancy (Gretton et al., 2012), Conditional Mean Embeddings (Song et al., 2009), Kernel Treatment Effects (Muandet et al., 2021), dimension-agnostic inference using cross U-statistic (Kim and Ramdas, 2023; Shekhar et al., 2022), and the normal asymptotic behaviour of the doubly robust estimator (Funk et al., 2011).
3.1 Maximum Mean Discrepancy (MMD) and Conditional Mean Embeddings
Let  $\mathcal{X}$  be a non-empty set and let  $\mathcal{H}$  be a Hilbert space of functions  $f:\mathcal{X}\to \mathbb{R}$  with inner product  $\langle \cdot ,\cdot \rangle_{\mathcal{H}}$  . A function  $k:\mathcal{X}\times \mathcal{X}\rightarrow \mathbb{R}$  is called a reproducing kernel of  $\mathcal{H}$  if (i)  $k(\cdot ,x)\in \mathcal{H}$  , for all  $x\in$ $\mathcal{X}$  ,(ii)  $\langle f,k(\cdot ,x)\rangle_{\mathcal{H}} = f(x)$  for all  $x\in \mathcal{X},f\in \mathcal{H}$  . If  $\mathcal{H}$  has a reproducing kernel, then it is called a Reproducing Kernel Hilbert Space (RKHS).
Building on a reproducing kernel  $k$  and a set of probability measures  $\mathcal{P}$ , the Kernel Mean Embedding (KME) maps distributions to elements in the corresponding Hilbert space as follows:
$$
\mu : \mathcal {P} \rightarrow \mathcal {H}, \quad \mathbb {P} \rightarrow \mu_ {\mathbb {P}} := \int k (\cdot , x) d \mathbb {P} (x).
$$
If the kernel  $k$  is "characteristic" (which is the case for frequently used kernels such as the RBF or Matern kernels), then  $\mu$  is injective. Conditional mean embeddings (Song et al., 2009) extend the concept of kernel mean embeddings to conditional distributions. Given two RKHS  $\mathcal{H}_{k_x},\mathcal{H}_{k_y}$  the conditional mean embedding operator is a Hilbert-Schmidt operator  $\mathcal{C}_{Y|X}:\mathcal{H}_{k_x}\to \mathcal{H}_{k_y}$  satisfying  $\mu_{Y|X = x} = \mathcal{C}_{Y|X}k_x(\cdot ,x)$ , where  $\mathcal{C}_{Y|X}\coloneqq \mathcal{C}_{YX}\mathcal{C}_{XX}^{-1}$ ,  $\mathcal{C}_{YX}\coloneqq \mathbb{E}_{Y,X}[k_y(\cdot ,Y)\otimes k_x(\cdot ,X)]$  and  $\mathcal{C}_{XX}\coloneqq \mathbb{E}_{X,X}[k_x(\cdot ,X)\otimes k_x(\cdot ,X)]$ . Given a dataset  $\{\mathbf{x},\mathbf{y}\}$ , a sample estimator may be defined as
$$
\hat {\mathcal {C}} _ {Y \mid X} = \Phi_ {\mathbf {y}} ^ {T} \left(K _ {\mathbf {x x}} + \lambda I\right) ^ {- 1} \Phi_ {\mathbf {x}}, \tag {1}
$$
where  $\Phi_{\mathbf{y}} := [k_y(\cdot, y_1), \ldots, k_y(\cdot, y_n)]^T$ ,  $\Phi_{\mathbf{x}} := [k_x(\cdot, x_1), \ldots, k_x(\cdot, x_n)]^T$ ,  $K_{\mathbf{xx}} := \Phi_{\mathbf{x}}\Phi_{\mathbf{x}}^T$  denotes the Gram matrix, and  $\lambda$  is a regularization parameter.
Building on the KME, Gretton et al. (2012) introduced the kernel maximum mean discrepancy (MMD). Given two distributions  $\mathbb{P}$  and  $\mathbb{Q}$  and a kernel  $k$ , the MMD is defined as the largest difference in expectations over functions in the unit ball of the respective RKHS:
$$
\operatorname {M M D} (\mathbb {P}, \mathbb {Q}) = \sup  _ {f \in \mathcal {H}} \mathbb {E} _ {X \sim \mathbb {P}} [ f (X) ] - \mathbb {E} _ {X ^ {\prime} \sim \mathbb {Q}} [ f (X ^ {\prime}) ] = | | \mu_ {\mathbb {P}} - \mu_ {\mathbb {Q}} | | _ {\mathcal {H}}.
$$
It can be shown (Gretton et al., 2012) that  $\mathrm{MMD}^2 (\mathbb{P},\mathbb{Q}) = ||\mu_{\mathbb{P}} - \mu_{\mathbb{Q}}||_{\mathcal{H}}^2$ . If  $k$  is characteristic, then  $\mathrm{MMD}(\mathbb{P},\mathbb{Q}) = 0$  if and only if  $\mathbb{P} = \mathbb{Q}$ . Given two samples drawn from  $\mathbb{P}$  and  $\mathbb{Q}$ , the MMD between the empirical distributions may be used to test the null hypothesis  $H_0:\mathbb{P} = \mathbb{Q}$  against  $H_{1}:\mathbb{P}\neq \mathbb{Q}$ . However, this statistic is a degenerate two-sample U-statistic under the null, thus one cannot analytically calculate the critical values. Consequently, a permutation-based resampling approach is widely used in practice (Gretton et al., 2012).
3.2 Kernel Treatment Effect: A distributional kernel-based treatment effect test
Based on the MMD and the potential outcomes framework, Kernel Treatment Effects (KTE) were introduced in Muandet et al. (2021) for testing distributional treatment effects in experimental settings (i.e. with known propensity scores).
Let  $(X_{i},A_{i},Y_{i})_{i = 1}^{n}\sim (X,A,Y)$  such that (i) (consistency)  $Y = A Y_{1}^{*} - (1 - A)Y_{0}^{*}$ , (ii) no unmeasured confounding and overlap assumptions hold. The KTE considers the MMD between  $Y_{0}^{*}$  and  $Y_{1}^{*}$  to test  $H_0:\mathbb{P}_{Y_0^*} = \mathbb{P}_{Y_1^*}$  against  $H_0:\mathbb{P}_{Y_0^*}\neq \mathbb{P}_{Y_1^*}$ . They define
$$
\widehat {\mathrm {K T E}} ^ {2} = \left| \left| \hat {\mu} _ {Y _ {1} ^ {*}} - \hat {\mu} _ {Y _ {0} ^ {*}} \right| \right| ^ {2},
$$
$$
\hat {\mu} _ {Y _ {1} ^ {*}} := \frac {1}{n} \sum_ {i = 1} ^ {n} \frac {A _ {i} k (\cdot , Y _ {i})}{\pi (X _ {i})}, \quad \hat {\mu} _ {Y _ {0} ^ {*}} := \frac {1}{n} \sum_ {i = 1} ^ {n} \frac {(1 - A _ {i}) k (\cdot , Y _ {i})}{1 - \pi (X _ {i})}.
$$
Alternatively, we may also define an unbiased version of it. Again, under the null, these are degenerate two-sample U-statistics. Hence, Muandet et al. (2021) proposed a permutation-based approach for thresholding.
The KTE considers the MMD between the mean embeddings of the counterfactual distributions. This is the cornerstone of the proposed distributional test, as will be exhibited in Section 4, which we extend in a doubly-robust manner to observational settings.
3.3 Permutation-free inference using cross U-statistics
The previously mentioned permutation-based approach for obtaining the threshold for the MMD statistic (and hence the KTE statistic) comes with finite-sample validity (Gretton et al., 2012). The number of permutations  $B$  used to find the empirical p-values generally varies from 100 to 1000. Consequently, the computational cost of finding a suitable threshold for the statistic is at least  $O(Bn^{2})$ . Such computational cost reduces the applicability of the approach, especially when time or computational resources are limited.
Driven by developing dimension-agnostic inference tools, Kim and Ramdas (2023) presented a permutation-free approach to test null hypotheses of the form  $H_0: \mu = 0$  against  $H_1: \mu \neq 0$ , where  $\mu$  is the mean embedding of a distribution  $\mathbb{P}$ , based on the idea of sample splitting. If  $X_1, \ldots, X_{2n} \sim \mathbb{P}$ , the usual degenerate V-statistic considers
$$
\widehat {\mathrm {M M D}} = \left\| \hat {\mu} _ {2 n} \right\| ^ {2},
$$
where  $\hat{\mu}_{2n} = \frac{1}{2n}\sum_{i=1}^{2n}k(\cdot,X_i)$  (or the similar unbiased version). Kim and Ramdas (2023) proposed to split the data in two and study
$$
\widehat {\mathbf {x M M D}} = \left\langle \hat {\mu} _ {n} ^ {A}, \hat {\mu} _ {n} ^ {B} \right\rangle ,
$$
where  $\hat{\mu}_n^A = \frac{1}{n}\sum_{i = 1}^n k(\cdot ,X_i),\hat{\mu}_n^B = \frac{1}{n}\sum_{j = n + 1}^{2n}k(\cdot ,X_j)$ . Denoting  $U_{i} = \langle k(\cdot ,X_{i}),\hat{\mu}_{n}^{B}\rangle$ , we have that
$$
\widehat {\mathbf {x M M D}} = \frac {1}{n} \sum_ {i = 1} ^ {n} U _ {i}.
$$
Under the null and some mild assumptions on the embeddings, Kim and Ramdas (2023, Theorem 4.2) proved that
$$
\widehat {x \mathrm {M M D}} := \frac {\sqrt {n \bar {U}}}{\hat {S} _ {u}} \stackrel {d} {\to} N (0, 1),
$$
where  $\bar{U} = \frac{1}{n}\sum_{i=1}^{n}U_i$ ,  $\hat{S}_u^2 = \frac{1}{n}\sum_{i=1}^{n}(U_i - \bar{U})^2$ . Consequently,  $\widehat{\bar{x}\mathrm{MMD}}$  is the statistic considered and the null is rejected when  $\widehat{\bar{x}\mathrm{MMD}} > z_{1-\alpha}$ , where  $z_{1-\alpha}$  is the  $(1-\alpha)$ -quantile of  $N(0,1)$ . Such test avoids the need for computing the threshold so it reduces the computational cost by a factor of  $\frac{1}{B}$  and it is minimax rate optimal in the  $L^2$  distance and hence its power cannot be improved beyond a constant factor (Kim and Ramdas, 2023).
The permutation-free nature of cross U-statistic is key in the proposed distributional test. It will allow us to circumvent the need for training regressors and propensity scores repeatedly, while preserving theoretical guarantees.
3.4 Empirical mean asymptotic behavior of AIPW
The main property from AIPW estimators that will be exploited in our proposed distributional treatment effect is the asymptotic empirical mean behaviour of the estimator. We present sufficient conditions in the next theorem.
Theorem 3.1. Let  $f(x, a, y) = \left\{\frac{a}{\pi(x)} - \frac{1 - a}{1 - \pi(x)}\right\} \left\{y - \theta_{a}(x)\right\} + \theta_{1}(x) - \theta_{0}(x)$ , so that  $\psi = \mathbb{E}\{f(X, A, Y)\}$  is the average treatment effect. Suppose that
-  $\hat{f}$  is constructed from an independent sample or  $f$  and  $\hat{f}$  are contained in a Donsker class.
-  $||\hat{f} - f|| = o_{\mathbb{P}}(1)$ .
Suppose also that (by clipping)  $\mathbb{P}(\hat{\pi}\in [\epsilon ,1 - \epsilon ]) = 1$ . If  $||\hat{\pi} -\pi ||\sum_{a}||\hat{\theta}_{a} - \theta_{a}|| = o_{\mathbb{P}}(\frac{1}{\sqrt{n}})$ , then it follows that
$$
\hat {\psi} _ {A I P W} - \psi = (\mathbb {P} _ {n} - \mathbb {P}) f (X, A, Y) + o _ {\mathbb {P}} (\frac {1}{\sqrt {n}}),
$$
so it is root-  $n$  consistent and and asymptotically normal.
Note that the IPW estimator can be seen as an AIPW with  $\hat{\theta}_0(X) = \hat{\theta}_1(X) = 0$  almost surely. The IPW estimator is also asymptotically normal if  $\| \hat{\pi} -\pi \| = o_{\mathbb{P}}(\frac{1}{\sqrt{n}})$ . In experimental settings  $\| \hat{\pi} -\pi \| = 0$ , hence the root-n rate is achieved. Under certain conditions (e.g., consistent nuisance estimation at  $n^{-1 / 4}$  rates), the asymptotic variance of the AIPW estimator is minimized for  $\hat{\theta}_{1} = \theta_{1}$ ,  $\hat{\theta}_0 = \theta_0$ , thus the IPW estimator is generally dominated by the AIPW if  $\hat{\theta}_1, \hat{\theta}_0$  are consistent.
The idea exhibited in Theorem 3.1 will allow for using cross U-statistics in estimated mean embeddings, rather than the actual embeddings. Nonetheless, Theorem 3.1 applies to finite-dimensional outcomes  $Y$ . We state and prove the extension of Theorem 3.1 to Hilbert spaces in Appendix C, which will be needed to prove the main result of this work.
4 Main results
We are now ready to introduce the main result of the paper. Let  $Z \equiv (X, A, Y) \sim \mathbb{P}$  be such that  $Y = AY_{1}^{*} + (1 - A)Y_{0}^{*}$  and that both no unmeasured confounding and overlap assumptions hold. We denote the space of observations by  $\mathcal{Z} = \mathcal{X} \times \mathcal{A} \times \mathcal{Y}$ . We are given  $Z_{i} \equiv (X_{i}, A_{i}, Y_{i})_{i=1}^{2n} \sim (X, A, Y)$  and we wish to test  $H_{0}: P_{Y_{1}^{*}} = P_{Y_{0}^{*}}$  against  $H_{1}: P_{Y_{1}^{*}} \neq P_{Y_{0}^{*}}$ . Given characteristic kernel  $k$  i.e.  $k(y, \tilde{y}) = \langle k(\cdot, y), k(\cdot, \tilde{y}) \rangle$  with induced RKHS  $\mathcal{H}_{k}$ , we equivalently test  $H_{0}: \mathbb{E}[k(\cdot, Y_{1}^{*}) - k(\cdot, Y_{0}^{*})] = 0$ .
Under consistency, no unmeasured confounding, and overlap, we have
$$
\mathbb {E} \left[ k \left(\cdot , Y _ {1} ^ {*}\right) - k \left(\cdot , Y _ {0} ^ {*}\right) \right] = \mathbb {E} [ \phi (Z) ],
$$
$$
\phi (z) = \left\{\frac {a}{\pi (x)} - \frac {1 - a}{1 - \pi (x)} \right\} \left\{k (\cdot , y) - \beta_ {a} (x) \right\} + \beta_ {1} (x) - \beta_ {0} (x),
$$
$$
\pi (x) = \mathbb {E} [ A \mid X = x ], \quad \beta_ {a} (x) = \mathbb {E} [ k (\cdot , Y) \mid A = a, X = x ].
$$
Note the change in notation, from  $\theta_{a}$  to  $\beta_{a}$ , to emphasize that such regression functions are now  $\mathcal{H}_k$ -valued. Thus, we can equivalently test for  $H_0: \mathbb{E}[\phi(Z)] = 0$ . With this goal in mind, we denote  $\mathcal{D}_1 = (X_i, A_i, Y_i)_{i=1}^n$ ,  $\mathcal{D}_2 = (X_j, A_j, Y_j)_{j=n+1}^{2n}$  and define
$$
T _ {h} ^ {\dagger} := \frac {\sqrt {n} \bar {f} _ {h} ^ {\dagger}}{S _ {h} ^ {\dagger}} \tag {2}
$$
$$
f _ {h} ^ {\dagger} (Z _ {i}) = \frac {1}{n} \sum_ {j = n + 1} ^ {2 n} \langle \hat {\phi} ^ {(1)} (Z _ {i}), \hat {\phi} ^ {(2)} (Z _ {j}) \rangle , i \in [ n ].
$$
Above,  $\hat{\phi}^{(r)}(z)$  is the plug-in estimate of  $\phi (z)$  for  $r\in \{1,2\}$  using  $\hat{\pi}^{(r)}$  and  $\hat{\beta}_a^{(r)}$ , which approximate  $\pi$  and  $\beta_{a}$  respectively. Further,  $\bar{f}_h^\dagger$  and  $S_h^\dagger$  denote the empirical mean and standard error of  $f_h^\dagger$ :
$$
\bar {f} _ {h} ^ {\dagger} = \frac {1}{n} \sum_ {i = 1} ^ {n} f _ {h} ^ {\dagger} (Z _ {i}), \quad S _ {h} ^ {\dagger} = \sqrt {\frac {1}{n} \sum_ {i = 1} ^ {n} (f _ {h} ^ {\dagger} (Z _ {i}) - \bar {f} _ {h} ^ {\dagger}) ^ {2}}.
$$
The next theorem, which is the main result of the paper, establishes sufficient conditions for  $T_h^\dagger$  to present Gaussian asymptotic behavior. While the main idea relies on combining cross U-statistics and the asymptotic empirical mean-like behaviour of AIPW estimators, we highlight a number of technical challenges underpinning this result. The proof combines the central idea presented in Kim and Ramdas (2023) with a variety of techniques including causal inference results, functional data analysis, and kernel method concepts. Furthermore, additional work is needed to extend Theorem 3.1 to  $\mathcal{H}_k$ -valued outcomes. Donsker classes are only defined for finite dimensional outcomes; in the  $\mathcal{H}_k$ -valued scenario, we ought to refer to asymptotically equicontinuous empirical processes (Park and Muandet, 2023) and Glivenko-Cantelli classes. We refer the reader to Appendix C for a presentation of such concepts, clarification of the norms used, and the proof of the theorem.
Theorem 4.1. Let  $k$  be a kernel that induces a separable RKHS and  $\mathbb{P}_{Y_0^*}, \mathbb{P}_{Y_1^*}$  be two distributions. Suppose that (i)  $\mathbb{E}[\phi(Z)] = 0$ , (ii)  $\mathbb{E}\left[\langle \phi(Z_1),\phi(Z_2)\rangle^2\right] > 0$ , (iii)  $\mathbb{E}\left[\| \phi(Z)\|_{\mathcal{H}}^4\right]$  is finite. For  $r \in \{1,2\}$ , suppose that (iv)  $\hat{\phi}^{(r)}$  is constructed independently from  $\mathcal{D}_r$  or (v) the empirical process of  $\hat{\phi}^{(r)}$  is asymptotically equicontinuous at  $\phi$  and  $\| \hat{\phi}^{(r)}\|_{\mathcal{H}}^2$  belongs to a Glivenko-Cantelli class. If it also holds that (vi)  $||\hat{\phi}^{(r)} - \phi|| = o_{\mathbb{P}}(1)$ , (vii)  $\mathbb{P}(\hat{\pi}^{(r)} \in [\epsilon,1 - \epsilon]) = 1$ , and
$$
\left| \left| \hat {\pi} ^ {(r)} - \pi \right| \right| \sum_ {a} \left| \left| \hat {\beta} _ {a} ^ {(r)} - \beta_ {a} \right| \right| = o _ {\mathbb {P}} \left(\frac {1}{\sqrt {n}}\right) \tag {3}
$$
for  $r\in \{1,2\}$ , then it follows that
$$
T _ {h} ^ {\dagger} \xrightarrow {d} N (0, 1).
$$
We would like to highlight the mildness of the assumptions of Theorem 4.1. The separability of the RKHS is achieved for any continuous kernel on separable  $\mathcal{V}$  (Hein and Bousquet, 2004). Assumption (i) is always attained under the null hypothesis (it is precisely the null hypothesis). Assumption (ii) prevents  $\phi(Z)$  from being constant. In such degenerate case,  $S_h^\dagger = 0$  thus  $T_h^\dagger$  is not even well-defined. Assumption (iii) is the more restrictive out of the first three assumptions, inherited from the use of Lyapunov's CLT in the proof. However, we note that this condition is immediately satisfied under frequently used kernels. For instance, under bounded kernels (for example the common Gaussian and Laplace kernels) such that  $\|k(\cdot,Y)\|_{\mathcal{H}} \leq M$ , we have that
$$
\| \beta_ {a} (x) \| _ {\mathcal {H}} = \| \mathbb {E} [ k (\cdot , Y) | A = a, X = x ] \| _ {\mathcal {H}} \leq \mathbb {E} [ \| k (\cdot , Y) \| _ {\mathcal {H}} | A = a, X = x ] \leq M,
$$
hence