On the role of surrogates in the efficient estimation of treatment effects with limited outcome data
Nathan Kallus Xiaojie Mao
stat.ML cs.LG stat.ME

Radiation from the impact of broad-line region clouds onto AGN accretion disks
    Jordan Williamson
    December 30, 2023
==============================================================================





In many investigations, the primary outcome of interest is difficult or expensive to collect. Examples include long-term health effects of medical interventions, measurements requiring expensive testing or follow-up, and outcomes only measurable on small panels as in marketing. This reduces effective sample sizes for estimating the average treatment effect (ATE). However, there is often an abundance of observations on surrogate outcomes not of primary interest, such as short-term health effects or online-ad click-through. We study the role of such surrogate observations in the efficient estimation of treatment effects. To quantify their value, we derive the semiparametric efficiency bounds on ATE estimation with and without the presence of surrogates and several intermediary settings. The difference between these characterize the efficiency gains from optimally leveraging surrogates. We study two regimes: when the number of surrogate observations is comparable to primary-outcome observations and when the former dominates the latter. We take an agnostic missing-data approach circumventing strong surrogate conditions previously assumed. To leverage surrogates' efficiency gains, we develop efficient ATE estimation and inference based on flexible machine-learning estimates of nuisance functions appearing in the influence functions we derive. We empirically demonstrate the gains by studying the long-term earnings effect of job training.



Keywords: 
Surrogate Observations, Causal Inference, Average Treatment Effect, Semiparametric Efficiency, Double Robustness.



§ INTRODUCTION

In many causal inference applications, it may be expensive, inconvenient or infeasible to measure the outcome of primary interest. 
Nevertheless, some auxiliary variables that are faster or easier to measure may be available. 
In clinical trials for AIDS treatment, the primary outcome is often mortality, which may take years of follow-up to fully reveal. 
But clinically relevant biomarkers like viral loads or CD4 counts can be measured quite rapidly <cit.>.  
In comparative effectiveness research for long-term impact of therapies, , long-term quality of life measures, many patients may drop-out so their responses are missing. But short-term outcomes, , responses shortly after the therapy, may be well recorded <cit.>. 
In program evaluation for addiction prevention projects, accurately measuring the primary outcome, , smoking behavior, may require costly chemical analysis of saliva samples for the presence of cotinine, and thus are available for only a limited number of participants. Yet self-report data are relatively inexpensive to collect <cit.>. 
In offline conversion analysis, we wish to assess the effect of a digital marketing campaign on visitation to a brick-and-mortar location. And, while we can only observe visitation for individuals for whom we have cellphone geolocation data and who we can match to ad identifiers, we can observe digital ad clicks for all units.
We refer to these easy-to-obtain auxiliary variables as surrogate outcomes or simply surrogates, which are often informative about or correlate with the primary outcome of interest.

There has been considerable interest in using surrogates as a replacement for the missing primary outcome to reduce data collection costs in causal inference. For example, the U.S. Food and Drug Administration (FDA) launched the Accelerated Approval Program to allow for early approval of drugs based on a clinically relevant surrogates, aiming to speed up clinical trials for drug approval <cit.>. 
This program is spurred by the urgent need to determine the efficacy of new drugs quickly and economically. 
As stated by the National Center for Advancing Translational Sciences (NCATS) of the U.S. National Institutes of Health, many thousands of diseases known to affect humans do not have any approved treatment yet;  meanwhile, a novel drug can “take 10 to 15 years and more than $2 billion to develop” <cit.>.
Therefore, accelerating drug approval is of great value and urgency to both pharmaceutical companies and patients.

Using surrogates that can be measured more easily provides a promising way toward this goal.

However, one major challenge is that surrogates may not be perfectly indicative of the primary outcome, so a misuse may lead to severe or even disastrous consequences. 
For example, three drugs (encainide, flecainide, and moricizine) were approved by FDA based on early success of supressing ventricular arrhythmia (surrogate), but in later follow-up trials the drugs alarmingly increased mortality (primary outcome) <cit.>. 
It was estimated that the use of these drugs resulted in 50,000 excess deaths in the U.S. <cit.>.
To resolve these problems, a wide variety of criteria have been proposed to ensure that it is adequate to base causal inference solely on the surrogate without observing the primary outcome at all.
 However, using these criteria to search for a valid surrogate to replace the primary outcome is still extremely challenging, since the criteria impose stringent assumptions that may 
 
 often be 
 violated in practice (see Related Literature in <Ref> and <Ref>). For example, the “statistical surrogate”  condition <cit.> requires the primary outcome to be conditionally independent of the treatment given surrogates, , surrogates must fully explain away the dependence of the outcome on the intervention meant to affect it. 
Not only does this condition require full mediation of the treatment effect, it is also easily invalidated if there is any other common cause of both surrogates and the primary outcome, which is often unavoidable even in ideal randomized trial settings (see <Ref>). 
Thus this surrogate condition, and similarly many other criteria that render surrogates perfect replacements for primary outcomes, are very 
prone to violation
in practice. 


In this paper, we view surrogates as supplements instead of replacements for the primary outcome.
 We  consider combining surrogates with the primary outcome, and investigate how this proposal can improve the efficiency of treatment effect estimation. 
Such combination is possible because in practice paired observations of both the primary outcome and surrogates are often available for at least some units.
By incorporating a limited number of primary-outcome observations, we can avoid the aforementioned problems resulting from relying on surrogates alone, and circumvent stringent surrogate conditions. 
Instead, we only assume standard causal inference assumptions and a typical missing data assumption that the primary outcome is missing (conditionally) at random (MAR), , any interdependence between the primary outcome value and whether it is observed or not may be explained by other observed variables (, pre-treatment covariates, treatment, surrogates).
Similar missingness conditions are also commonly assumed in previous literature that combine different datasets <cit.>. 
Under only these standard assumptions, and in particular no overly restrictive surrogate conditions,
we aim to investigate the role of surrogates in estimating treatment effects when the primary-outcome observations are limited. 

We first study the possible extent of benefit achievable by leveraging surrogate information.
Using the theory of semiparametric efficiency,  
we derive the efficiency lower bound of estimating the average treatment effect (ATE) on the primary outcome (<Ref>). This lower bound characterizes the fundamental statistical limit in estimating ATE under our assumptions, in that it is the best possible precision of ATE estimation that can be asymptotically achieved by any regular estimator. 
By comparing the efficiency lower bounds both with and without the presence of surrogates, and bounds in several intermediary settings (<Ref>), we precisely quantify the efficiency gains from surrogates, namely, the benefit of surrogates in terms of allowing us to estimate treatment effects up to the same precision with fewer observations of the primary outcome. 

We find that using surrogates is particularly advantageous when (i) the primary outcome is missing for a large number of units, and (ii) the surrogates are reasonably predictive of the primary outcome, in that they can account for large variations of the primary outcome that cannot be explained by the pre-treatment covariates, but they need not determine them exactly or render them independent of treatment.
These theoretical results provide insightful guidelines for understanding when surrogates can yield significant benefits.
Moreover, we show that essentially the same efficiency lower bound (under appropriate reformulation) reigns across two different regimes: when the size of the unlabeled data is comparable to the size of the labeled data (<Ref>), and when the former is much larger than the latter (<Ref>). 
In the second regime, the commonly assumed overlap condition (<Ref> condition (<ref>)) fails and the efficiency analysis under MAR setting becomes more challenging.
Our paper tackles this very practical setting when enormous amounts of cheap unlabeled data may be available.

We further propose an ATE estimator that can optimally leverage the efficiency gains from surrogates and achieves the efficiency lower bound. 
The proposed estimator involves some nuisance parameters that are of no intrinsic interest but need to be estimated first. By employing a cross-fitting technique <cit.>, we can allow for any flexible machine learning estimators to be used for the nuisance parameters as long as they satisfy some generic convergence rate conditions.
We show that the proposed estimator converges to the true ATE value, even if only some but not all nuisance parameters are consistently estimated (<Ref>). 
If all nuisance parameters are indeed consistently estimated under generic rate conditions, then the proposed estimator is asymptotically normal centered at the true ATE value and its asymptotic variance attains the efficiency lower bound (<Ref>).
Furthermore, we construct asymptotically valid confidence intervals based on a simple plug-in estimator for the asymptotic variance of our ATE estimator (<Ref>). 
In summary, we propose an ATE estimator that can leverage the power of flexible machine learning estimators for nuisance estimation, is robust to nuisance estimation errors, achieves full asymptotic efficiency in leveraging surrogate information, and may be combined with easy-to-use inference. 

Our paper is organized as follows. In <Ref>, we set up the problem of treatment effect estimation with surrogates when the primary outcome is not fully observed, and introduce our notation. 
In <Ref>, we derive the efficiency lower bound for ATE estimation in our setting, and compare it with bounds in other benchmark settings to characterize the efficiency gains from surrogates. 
We then construct an asymptotically efficient estimator and prove its asymptotic properties in <Ref>. 
In <Ref>, we extend the efficiency and estimation results to the setting where the amount of unlabeled data is much larger than amount of labeled data. We point out the efficiency and estimation results in this setting are consistent with those in 
<Ref>.
In <Ref>, we use our methods to study the effect of a job training intervention on earnings at a later follow up using data from a large-scale randomized controlled trial <cit.> and we demonstrate the gains due to employing surrogates and due to our methods.
We provide concluding remarks in <Ref>. In <Ref>, we provide supplementary discussions about the “statistical surrogate” condition, the connection of our work to some previous literature, and additional details that 
expand on our results from <Ref>, respectively. All proofs are relegated to <Ref>. 
 



 §.§ Problem Setup

Let T ∈{0, 1} denote a treatment indicator variable  (, T = 1 means being treated with a therapy of interest, and T = 0 means control), X  ∈𝒳⊆ℝ^d_x denote baseline covariates  measured prior to treatment (, patients' demographic characteristics and health measurements before treatment), and Y ∈ℝ denotes the  outcome variable of primary interest (, patients' health outcome after treatment). Following the Neyman-Rubin potential outcome framework <cit.>, we assume the existence of two potential outcomes Y(1), Y(0) corresponding to the outcomes that would have been realized under each treatment option. We assume that the actual observed outcome is the potential outcome corresponding to the actual treatment, , Y = Y(T), which encapsulates the non-interference and
consistency assumptions in causal inference <cit.>. Our goal is to estimate the average treatment effect (ATE):

    δ^* = ξ_1^* - ξ_0^*,  where  ξ_t^*= [Y(t)] for t=0,1.

If we could observe (X, T, Y) for all units, then we could estimate the ATE  by many existing methods <cit.>.


In this paper, we consider 
a more challenging 
setting where the primary outcome Y cannot be observed for all units, due to long follow-up, drop-out, budget constraints, etc. Nonetheless, we can observe for all units some surrogates S ∈𝒮⊆ℝ^d_s (, intermediate outcomes) that may be informative about the primary outcome Y (, a long-term outcome). Since surrogates are measured after the treatment assignment, they may also be affected by the treatment. Thus we hypothesize the existence of two potential surrogate outcomes S(1), S(0) analogously, and assume S = S(T). We use R ∈{0, 1} to denote the indicator of whether the primary outcome Y is observed.  In summary, we can observe a labeled subset {(X_i, T_i, S_i, Y_i, R_i = 1): i ∈ℐ^l}, and an unlabeled subset {(X_i, T_i, S_i, Y_i = NA, R_i = 0): i ∈ℐ^u}, where NA stands for “not available” (missing value), and ℐ^l and ℐ^u are the index sets for labeled data and unlabeled data respectively. We denote N_l = |ℐ^l|, N_u = |ℐ^u| as the corresponding sample sizes for these two datasets, and N = N_l + N_u as the total sample size. We represent the ith data point as W_i = (X_i, T_i, S_i, Y_i, R_i), and assume that each data point (in both labeled subset and unlabeled subset) is given by coarsening an independent and identically distributed (i.i.d) draw from a population W^*=(X, T, S(0), S(1), Y(0), Y(1), R) with distribution characterized by a probability measure ^*. In particular, the coarsening map transforming such draws to the data we actually observe is given by 

    𝒞:(X, T, S(0), S(1), Y(0), Y(1), R)↦(X,T,S(T),R× Y(T)+(1-R)×(NA),R).
 We let  denote the distribution on W=𝒞(W^*) induced by ^*. Depending on the context, 
we may use  to denote expectation with respect to either ^* or . 



[Unconfoundedness]
For t = 0, 1,

    (Y(t), S(t)) ⊥ T    | X .



In <Ref>, we first assume that the treatment assignment is unconfounded in the combined population of labelled and unlabelled data. 
In <Ref>, we will show that this condition is guaranteed if the treatment is unconfounded in both the labelled and unlabelled  subpopulation, separately, when additional missing-at-random assumptions are imposed. 
This condition requires that X include all potential confounders that can affect the primary outcome and treatment simultaneously, or the surrogate and treatment simultaneously.
It is trivially satisfied by design in clinical trials where the treatment T is assigned totally at random. 



[Missing at random]
For t = 0, 1, 

    R ⊥ Y(t) | X, S(t), T = t.



In <Ref>, we assume that the primary outcome is missing (conditionally) at random (MAR), , the indicator R depends on only observed variables, including pre-treatment covariates X, the surrogates S, and the treatment T. 
This condition guarantees that the distribution of the primary outcome on the labeled data and unlabeled data are comparable after accounting for the observed variables, so that we can use the labeled data to infer information about the missing primary outcome in the unlabeled data.  
This condition is considerably weaker than the  missing completely at random (MCAR) condition typically assumed in previous semi-supervised inference literature <cit.>, since MCAR does not allow the missingness of the primary outcome to depend on any other variable. 
<Ref> may be satisfied by design in a two-phase sampling scheme <cit.>: in the first phase, relatively cheap measurements of T, X, S are available for all units, and in the second phase, expensive measurements of the primary outcome Y are collected for a validation subsample selected according to a selection rule based on variables measured in the first phase. 
For example, we may want to oversample units who self-report no-smoking behavior for further chemical analysis, if we suspect more misreporting in this subpopulation.




We next define some important quantities for ATE estimation. 
We first define the regression function  of the primary outcome in the labeled dataset, conditional on treatment, covariates, and surrogates, and also the projection of  onto the whole population, conditional on only treatment and covariates:

    (t, x, s)    = [Y | T = t, X = x, S = s, R = 1] ,
    (t, x)    = [(T, X, S) | T = t, X = x].


We also define the propensity scores for treatment and labeling respectively: 
    (x)    = (T = 1 | X = x),  (t, x, s) = (R = 1 | T = t, X = x, S = s).

Although these quantities are useful for estimating the ATE, they are of no intrinsic interest by themselves, so we refer to them as nuisance parameters. We let η^* = (, , , ) be the collection of the true nuisances. We further assume the following strict overlap assumption for the two propensity scores. 

[Strict Overlap]

There exist ϵ∈ (0, 1/2) such that almost surely we have 

    ϵ   ≤(X) ≤ 1 - ϵ,  
    ϵ   ≤(t, X, S(t)) ≤  1,  for  t = 0, 1.



This assumption states that units with any given values of the conditioning variables above have at least probability of ϵ to receive each treatment option, and to get their primary outcome measured.  
This overlap assumption is common in causal inference and missing data literature <cit.>. 
Note condition (<ref>) implies that (R = 1) ≥ϵ, so the size of the unlabeled data is necessarily comparable to the size of the labeled data, , N_u ≍ N_l (unless all data is labeled).
In <Ref>, we will relax this condition and consider the setting where enormous cheap unlabeled data are available so that N_u ≫ N_l. 

Below we show identification of the the  ATE parameter δ^*.

If <Ref> hold, then 

    δ^* 
       = Y | T = 1, R = 1, X, S| X, T = 1
                        - Y | T = 0, R = 1, X, S| X, T = 0.



In this paper, we focus on the efficient estimation of ATE  (, δ^* in <ref>) when the primary outcome Y is missing for many units while surrogates S can be fully observed for all. Notably, we only assume <Ref> (and some straightforward variants) that are very typical in causal inference and missing data literature. 
In particular, we do not assume any strong surrogate conditions such as the “statistical surrogate” condition, Y ⊥ T | S, X, R= 1, which may impose  restrictions that can easily be violated in practice (see <Ref> and <Ref> for more discussion).



  
*Notation. We use O, o, O_p, o_p to denote the nonstochastic and stochastic asymptotic orders, respectively. For nonstochastic sequences a_N≥0 and b_N>0, a_N = O(b_N) if lim sup_N →∞ a_N/b_N < ∞ and a_N = o(b_N) if lim_N →∞ a_N/b_N = 0. For a random variable sequence Z_N, we denote Z_N = O_p(a_N) if for any positive constant ε, there exists a finite positive constant M such that (|Z_N/a_N| > M) < ε, and we denote Z_N = o_p(a_N) if for any positive constant ε, (|Z_N/a_N| > ε) → 0 as N →∞.
We also use the notation ≍ and ≫, ≪ for asymptotic orders (both stochastic and nonstochastic). For example, for nonstochastic asymptotic order, a_N ≍ b_N if a_N = O(b_N) and b_N = O(a_N), a_N ≫ b_N if b_N/a_N = o(1), and a_N ≪ b_N if a_N/b_N = o(1).
For an appropriately measurable and integrable function f, we use f, f_p, f_∞ to denote the L_2, L_p and L_∞ norms with respective to the measure : f = {[f^2(W)]}^1/2, f_p = {[|f(W)|^p]}^1/p, and f_∞ = inf{c ≥ 0: (|f(W)| ≤ c) = 1}.
 
 Throughout this paper, we use ·^* to denote unknown population quantities like δ^* and η^*, and use ·̂   denote estimators, , δ̂.






 §.§ Related Literature



  
*Causal inference with surrogates. Many different surrogate criteria have been proposed to guarantee that the treatment effect  on a surrogate will reliably predict the treatment effect on the primary outcome. 
The statistical surrogate criterion proposed by <cit.> was the first such criterion, which requires the primary outcome to be conditionally independent of the treatment, given the surrogate. 
Since then, many other criteria have been proposed to improve upon the statistical surrogate criterion, such as the principal surrogate criterion <cit.>, strong surrogate
criterion <cit.>, consistent surrogate criterion <cit.>, among many others. 
However, almost all of these criteria involve quantities that cannot be identified from the observed data, so they are unverifiable in practice. 
Moreover, many of them can easily run into a logical paradox  described by 
<cit.>.
See <cit.> for a comprehensive review of surrogate criteria and <Ref> for a detailed discussion about the statistical surrogate condition.




While the literature above mostly focus on a single surrogate,
<cit.> propose to estimate  transformations of multiple surrogates to optimally approximate the primary outcome using labelled experimental data.
Their optimal transformations can avoid the surrogate paradox described in <cit.>. 
<cit.>  consider identifying and estimating the average treatment effect with multiple surrogates in the setting where the primary outcome cannot be observed simultaneously with treatment variable and are instead observed in two separate datasets, connected only by the surrogates and covariates. 
This setting is practically very challenging, since the two datasets have no complete observations at all, with the primary outcome missing in one dataset and treatment missing in the other. 
To fuse these two incomplete datasets and have hope of relating the effect of treatments on downstream outcomes, they have to assume the statistical surrogate condition, which, however, may be too strong in practice as it requires both full mediation of treatment effect by the surrogates and the absence of any other common cause of primary outcome and surrogate (see <ref>).
<cit.> use surrogates to combine experimental data with short-term observations and confounded observational data with long-term observations, the former using a latent unconfoundedness assumption and latter using multiple sequential surrogates as proxy variables. All of these focus on leveraging surrogates to enable identification, rather than improve efficiency in already-identified settings, as we do here.


<cit.> study efficient ATE estimation when combining a small number of primary-outcome observations with many observations of the surrogates, without assuming any surrogate criteria like those mentioned above. 
Their setting is closest to ours, except that they focus on the case when the unlabeled dataset is much larger than the labeled dataset, , N_u ≫ N_l and they assume that the primary outcome is MCAR. In contrast, our paper studies both N_u ≫ N_l and N_u ≍ N_l and considers a more general MAR setting.
By studying both N_u ≫ N_l and N_u ≍ N_l, we discover that essentially the same efficiency lower bound governs both regimes. 
Moreover, <cit.> consider certain specialized estimators based on parametric regressions and kernel smoothing, 
while 
our proposed estimator can leverage flexible machine learning nuisance estimation.
See <Ref> for a more detailed comparison of our work with <cit.>.



 


  
*Semi-supervised inference. Our paper is related to the growing body of  literature on parameter estimation and inference in the  semi-supervised setting where a small labeled dataset is enriched with a large unlabeled dataset. 
A stream of research has investigated how to the use unlabeled data to aid in the estimation of a wide variety of parameters, including regression coefficients <cit.>, population mean and average treatment effect <cit.>, 
quantiles and quantile treatment effect <cit.>, 
performance measures of a given classifier 

<cit.>, covariance <cit.>, etc. 
These papers typically propose estimators for a finite dimensional target parameter and study their asymptotic performance, which supplement extensive literature on semi-supervised learning of predictive models <cit.>.
Nearly all of these literature implicitly or explicitly assume that labels are MCAR. 
Our paper relaxes this assumption by allowing the labeling process to depend on pre-treatment covariates, the treatment, and even the surrogates, which is more realistic if the labeling process is not completely controlled by design. Moreover, while we study the intermediate setting of partial outcome labeling, our paper also studies and focuses on the use of surrogates as a source of extra information.
Interestingly, when viewing the surrogates in our paper as empty, our results also recover results in existing semi-supervised inference literature, for example, <cit.>. See <Ref> for details.  




  
*Measurement error problems with a validation sample. 
We can also view our problem as a measurement error problem: abundant  mismeasurements of the primary outcome (, the surrogate observations) are available, while accurate measurements (, the primary outcomes itself) are  observed only on a small validation sample (, the labeled dataset). 
In similar data settings, a variety of methods have been proposed to leverage observations with measurement noise to improve the efficiency of estimating regression coefficients <cit.> or solutions to estimation equations <cit.>, when either the outcome or covariates of interest are mismeasured.
In particular, some of these works <cit.> employ semiparametric methods  that do not impose restrictive assumptions on the measurement errors, and 
use the validation sample to nonparametrically estimate 
the relationship between surrogate variables and the primary variable. 
Our paper follows the same principle, but we allow for arbitrary machine learning nuisance estimators under only generic rate conditions.
Some literature also cast this type of problem as a missing data problem where the variables of primary interest are missing for all units not in the validation sample, and they study the semiparametric efficiency lower bound and efficient estimator for parameters in regression problems <cit.>.
Our paper builds on the missing data framework to study the efficiency of estimating treatment effects in presence of surrogates.
Moreover, in contrast to the missing data literature that commonly assume the proportion of complete observations to be bounded away from 0, our paper allows the complete-case proportion to vanish to 0 in order to model the setting with enormous amounts of unlabeled surrogate data. 


















§ EFFICIENCY ANALYSIS


In this section we derive the efficiency lower bounds and efficient influence functions in a sequence of models ranging from no surrogate observations to full outcome observations on all data points, crucially including our primary setting of interest as a practical middle ground (see <ref>). This serves to quantify both the information gain from surrogate observations relative to no surrogate observations and the gap remaining relative to full outcome observations. Later, this will also serve as motivation for the construction of our efficient ATE estimator.



 §.§ Efficiency Analysis in the Presence of Surrogates


We first derive the efficiency lower bound for ATE estimation in our primary setting of interest as described in <ref>. 


Let ℳ be the set of all distributions  on W induced by the coarsening map 𝒞 in <ref> applied to any distribution ^* on W^* satisfying <ref>. 
The semiparametric efficiency lower bound for δ^* under  model ℳ is V^* = [ψ^2(W; δ^*, η^*) ] where 

    ψ(W; δ^*, η^*) =   (1, X) - (0, X) - δ^* 
           + TR/(X)(1, X, S)(Y - (1, X, S)) - (1 - T)R/(1 - (X))(0, X, S)(Y - (0, X, S)) 
           + T/(X)((1, X, S) - (1 , X)) - 1 - T/1 - (X)((0, X, S) - (0 , X)).
 


<Ref> reveals the fundamental statistical limit in estimating δ^* with surrogates under <Ref>: for any regular estimator δ̂, the variance of the limiting distribution of √(N) (δ̂ - δ^*) must be no smaller than V^*.
In other words, V^* is the best possible precision we can aim to achieve asymptotically among all regular estimators. 
The function ψ(W; δ^*, η^*) is the efficient influence function for δ^*, which will be used to construct efficient estimators for δ^* in <Ref>.

Notably, the efficiency lower bound here corresponds to the model that only assumes <Ref>, but not any other strong surrogate condition.
To study the role of surrogates in the efficient estimation of ATE, 
we next consider the efficiency bound in a few other settings. 



 §.§ Efficiency Analysis in Other Settings




To quantify the benefit of surrogates in estimating ATE, we compare the efficiency lower bounds in following different settings. 


 
  Setting I: no surrogate (<Ref>). We observe (X, T, Y) for R = 1 and observe (X, T) for R = 0;
 
  Setting II: surrogate only on labeled data (<Ref>).  We observe (X, T, S, Y) for R = 1 and observe (X, T) for R = 0;
 
  Setting III: surrogate on all data (<Ref>). We observe (X, T, S, Y) for R = 1 and observe (X, T, S) for R = 0;
 
  Setting IV: fully labeled data (<Ref>). We observe (X, T, S, Y) for all units.
  


From setting I to setting IV in <Ref>, more information is increasingly observed  . 
 Setting I corresponds to one extreme where no surrogates are observed at all, and setting IV corresponds to the other extreme where all variables (including the primary outcome) are always completely observed. 
In the intermediate setting II, we observe surrogates only for units whose primary outcome is already observed, and setting III corresponds to our primary problem setup in <Ref>, where surrogates are always observed.

Each of these settings can be described by different choices for the coarsening map 𝒞. To compare the efficiency gains of the additional information in each setting, we can consider the efficiency bound corresponding to the same ^* as different coarsening maps 𝒞 are applied, each corresponding to one of the above settings.
Each map induces a model given by the distributions  induced by all ^* that satisfy <ref>.
Crucially, we will need that in each setting we have identifiability,
meaning that if ^*' and ^* induce the same data distributions, '=, under 𝒞, then they also induce the same ATE, δ^*, so that δ^* is a valid function of . This ensures we are in fact considering the same estimand in each of the models. In our primary setting (, setting III), restricting to ^* to satisfy <ref> is enough to ensure identifiability. In settings I and II, since surrogates S are not observed for some units,
we need to further assume that whether the primary outcome is observed or not, , indicator variable R, does not depend on surrogates. 

[Missing at random, cont'd]
For t= 0, 1, R ⊥ S(t) | T, X. 


With this additional assumption, the ATE parameter is identifiable in all four settings, so we can  compare the efficiency of estimating the same ATE in these different settings. 

If <Ref> all hold, then the ATE parameter δ^* is identified in all four settings in <Ref>. 



In the following lemma, we summarize some additional implications of <Ref>.

 If <Ref> hold, then <Ref> holds if and only if 
 
    (Y(t), S(t)) ⊥ T | X, R = i,    i ∈0, 1.

Moreover, when <Ref> holds,  (t, x, s) =(t, x) (R = 1 | T  = t, X = x) and (t, x) = [Y | T = t, X = x,R = 1]. 
 

In <Ref>, <Ref>
shows that under the missing-at-random assumptions in <Ref>, 
the treatment 
unconfoundedness over the combined population of the labelled and unlabelled data in <Ref> is equivalent to unconfoundedness over the two v subpopulations respectively. 
Moreover, <Ref> shows that <Ref> can also simplify two nuisances that appear in the efficient influence function in <Ref>. 
This is very beneficial because the simplified nuisances are easier to estimate. 
For example, the nuisance function (t, x) = [Y | T = t, X = x,R = 1] can be directly estimated by running regressions. 
In contrast, estimating the nuisance function  in <Ref> requires first estimating another nuisance  in <Ref> and then further projecting the estimated nuisances.
 

In the following theorem, we derive efficiency lower bounds for ATE in the four settings in <Ref>. We impose <Ref> even in settings III and IV where it is not needed for identification, else the four settings would not be comparable.


Under <Ref>, the efficiency lower bounds for δ^* in setting j is V^*_j = [ψ^2_j(W; δ^*, η^*)] for j = I, …, IV, where 

    ψ_I(W; δ^*, η^*)    = ψ_II(W; δ^*, η^*) = (1, X) - (0, X) - δ^*  
                  + TR/(X)(1, X)(Y - (1, X)) - (1 - T)R/(1 - (X))(0, X)(Y - (0, X)), 
    ψ_III(W; δ^*, η^*) 
           = (1, X) - (0, X)  - δ^* 
       =+ TR/(X)(1, X)(Y - (1, X, S)) - (1 - T)R/(1 - (X))(0, X)(Y - (0, X, S)) 
       =+ T/(X)((1, X, S) - (1 , X)) - 1 - T/1 - (X)((0, X, S) - (0 , X)) ,  
    ψ_IV(W; δ^*, η^*) 
           =  (1, X) - (0, X) - δ^* + T/(X)(Y - (1, X)) - 1 - T/(1 - (X))(Y - (0, X)).



In <Ref>, we prove that the efficiency lower bound for setting III 
is identical to the bound in <Ref>, meaning that the additional <Ref> has no impact on the efficiency bound.
Moreover, we note that even though we have access to surrogates for at least a subset of units in setting II and IV, their efficiency lower bounds do not depend on surrogates S.  
This means that surrogates cannot improve the efficiency of ATE estimation if surrogates are observed only when the primary outcome is already observed. 
Indeed, for units whose primary outcome is already observed, surrogates can provide no extra information for ATE, especially considering that we do not restrict the relationship between surrogates and the primary outcome at all (, no parametric modeling assumptions).
In contrast, for units whose primary outcome is missing, the observed surrogates do provide extra information, because under <Ref>, we can learn the relationship between surrogates and the primary outcome based on the labeled data and extrapolate it to the unlabeled data to impute the missing primary outcome.
 

Suppose <Ref> hold. Then: 


  * The efficiency gain from observing the surrogates on all units is measured by 

    V_I^* - V^*_III   = [1 - (1, X)/(X)(1, X)[(1, X, S(1))| X] + 1 - (0, X)/(1 - (X))(0, X)[(0, X, S(0))| X]].


  * The information loss due to not fully observing the primary outcome 

is measured by 

    V^*_III  - V^*_IV = [1 - (1, X)/(X)(1, X)[Y(1) | X, S(1)] + 1 - (0, X)/(1 - (X))(0, X)[Y(0) | X, S(0)]].


  * Observing additional outcomes on the labeled data alone provides no improvement, that is, V_I^* = V^*_II.



<Ref> quantifies the  efficiency gain from optimally leveraging surrogates, and the efficiency gap to the ideal setting where the primary outcome is fully observed. 
It shows that the efficiency benefits of surrogates crucially depend on two factors: the predictiveness of the surrogates with respect to the primary outcome 
and the extent of missingness of the primary outcome.

Predictiveness of the surrogates. The efficiency gain due to surrogates (, V_I^* - V^*_III) positively depends on the terms [(t, X, S(t))| X] for t = 0, 1 , the variations of the primary outcome that can be explained by surrogates but not by pre-treatment covariates. Similarly, the efficiency loss compared to the ideal setting (, V^*_III  - V^*_IV) positively depends on [Y(t) | X, S(t)] for t = 0, 1, , the residual variations of the primary outcome that cannot be explained by either the surrogates or the pre-treatment covariates. 
This means that the more predictive the surrogates are, the more efficiency improvement can be achieved by leveraging the surrogates (, larger V_I^* - V^*_III), and the closer the efficiency bound is to the ideal limit with fully observed primary outcome (, smaller V_III^* - V^*_IV).   
In the extreme, if surrogates S can perfectly predict Y, , there exists a deterministic function that maps S to Y, then observing surrogates is equivalent to observing the primary outcome, and thus V_III^* = V^*_IV. Note this deterministic function need not be known.

Missingness of the primary outcome. Both quantities in <Ref> decrease with the labeling propensity scores (1, X) and (0, X). This means that when the primary outcome is less missing (, overall higher labeling propensity scores), the efficiency gains from additionally observing surrogates or the primary outcome both decrease. Indeed, if the primary outcome is already observed for most of the units, then the room for extra efficiency gain from observing surrogates (or, from observing more primary outcomes, for that matter) is small.  

The efficiency analysis above provides important guidelines on when leveraging surrogates can improve the efficiency of ATE estimation. It shows that surrogates are particularly beneficial for ATE estimation when (1) surrogates can account for large variations of the primary outcome that cannot be explained by the pre-treatment covariates, and (2) the primary outcome for a large number of units is missing. 




 §.§ Aside: Other Target Populations


In the above we considered our estimand to be the ATE on the whole population described by ^*. 
To identify this estimand, we require the  treatment assignment to be  unconfounded on the whole population (<Ref>). 
According to <Ref>, under the missing-at-random assumptions in <Ref>, the whole-population unconfoundedness <Ref> amounts to unconfoundedness on both the labelled (R = 1) and unlabelled (R = 0) subpopulations, separately. 

We could easily consider the ATE on other target populations, for example, [Y(1)-Y(0)| R=i] for either i=0,1, that is, the ATE on the unlabeled  or labeled  subpopulation. We may be interested in δ_1^* = [Y(1)-Y(0)| R=1] if, for example, the unlabeled data is collected from an auxiliary source to augment a small study already involving the population of interest. 
Or, we may be interested in δ_0^* = [Y(1)-Y(0)| R=0] if the unlabelled data are easier to collect and more representative of the population of interest. 
For these alternative estimands, we only need the treatment assignment to be unconfounded for the target population of interest.

When the target estimand is the ATE on the labelled population δ_1^*, we only need  unconfoundedness on the labelled population, ,  Y(t) ⊥ T | X, R = 1.
Moreover, we only need strict overlap assumption on the treatment assignment (namely, <Ref> in <Ref>). 
In particular, 
the missing-at-random assumptions in <Ref> are not required. 
In this case, the surrogates are not useful since we already fully observe the primary outcome in the labelled population of interest. 
The semiparametric efficiency analysis of δ_1^* immediately follow from existing literature such as <cit.>. 

When the target estimand is the ATE on the unlabelled population δ_0^*, we only need unconfoundedness on the unlabelled population.  
In the following theorem, we show the identification of δ^*_0 and its semiparametric efficiency bound. 


If Y(t), S(t)⊥ T | X, R = 0 and <Ref> hold, then 

    δ_0^* 
       = Y | S, X, T = 1, R = 1| X, T = 1, R = 0| R = 0
               - Y | S, X, T = 0, R = 1| X, T=0, R = 0| R = 0.

The corresponding semiparametric efficiency bound is V_0^* = ψ^2_0(W; δ_0^*), where 

    ψ_0(W; δ_0^*) 
          = 1-R/R=0_0(1, X) - _0(0, X) - δ_0^* + T-(0,X)/(0, X)(1-(0, X))(T, X, S) - _0(T, X)
       +R/R=0R=0| S, X, T/R=1| S, X, TT-(0,X)/(0, X)(1-(0, X))Y - (1, X, S),

and  
_0(t, x)= [(T, X, S) | X = x, T = t, R = 0], (0, X) = T=1| R=0, X.


Based on the efficient influence function in <Ref>, we can easily adapt our estimation method in <Ref> to construct efficient estimators for the parameter δ_0^*. 



§ TREATMENT EFFECT ESTIMATOR


In this section we develop our treatment effect estimator that efficiently leverages surrogates and achieves the bound derived in <ref>.
We first show how to construct the estimator in <Ref>. Then we establish the asymptotic guarantees for our estimator in <Ref>.




 §.§ Constructing an Efficient Estimator


Our analysis in <ref> not only provides the efficiency bound for ATE estimation, but also guides us directly in the construction of an efficient estimator.
In particular, <ref> suggests one hypothetical estimator if the nuisance parameters η^* = (, , , ) were known: specifically, the efficient influence function itself in <ref> gives the estimator δ̂_0 that solves the following estimating equation: 

    1/N∑_i = 1^N ψ(W_i; δ̂_0, η^*) = 0.

It is then easy to verify by the Central Limit Theorem (CLT) that 
√(N)(δ̂_0 - δ^*) d→𝒩(0, V^*), which validates the efficiency of δ̂_0.

However, in practice, we do not know the nuisance parameters, so the estimator δ̂_0 is infeasible. Instead, our approach will be to construct some nuisance parameter estimators η̂ = (, , , ) first, and then plug them into <ref> in place of η^*.
We could estimate η^* using parametric models (, linear regression, logistic regression), but this would risk model misspecification and lead to inconsistent estimates, let alone efficiency.
This is particularly a concern when either covariates X or surrogates S are rich, which should normally be regarded as a good thing as it can make <ref> more defensible as well as increase surrogates' predictiveness and hence the efficiency gains from surrogate observations.
Hence, we prefer flexible machine learning estimators 
 that avoid restrictive parametric  assumptions on the nuisance parameters to avoid misspecification error.
For example, estimating , amounts to learning conditional probabilities from binary classification data, and estimating , is essentially learning two regression functions. For both tasks many successful machine learning methods exist, such as random forest <cit.>, gradient boosting <cit.>, neural networks <cit.>, 



Although flexible machine learning estimators are less prone to model misspecification, we must be careful that their slow convergence and possible biases do not impact our estimator badly so that the resulting feasible ATE estimator is still root-N consistent and asymptotically normal, just like the hypothetical estimator in  <Ref>. 
Luckily, the efficient influence function we derive in <ref> has a special multiplicative bias structure that makes it naturally insensitive to errors in η^*.


For any η_0 = (e_0, r_0, μ̃_0, μ_0) such that e_0, r_0 satisfy <ref> and any δ, 

    ψ(W; δ, η_0)-ψ(W; δ, η^*)≤√(2)ϵ^-3/2e_0 - _2μ_0 - _2 + ϵ^-3r_0 - _2_0 - _2.





<Ref> suggest that replacing η^* with η̂ in <ref>, the estimate will remain consistent even if only one of  and  and one of  and  are consistent, as we will show in <Ref>. 
More crucially, <Ref> suggests that if all nuisance estimators are consistent but converge slowly, then the overall error in using η̂ in place of η^* in <ref> will converge as the product of the slow rates. If this product is faster than the O_p(N^-1/2) convergence of δ̂_0 itself, , each rate is o_p(N^-1/4), then our feasible estimator will asymptotically behave the same as the infeasible one up to leading orders (see <Ref> below for the  formal statements).
The property shown in <Ref> implies the Neyman orthogonality property that plays a central role in the recent debiased machine learning literature <cit.>. 

To construct our estimator, we further employ cross-fitting in nuisance estimation.
We divide the data into multiple folds, use data in all but one fold to estimate nuisances, and apply the estimated nuisance only to the hold-out fold. 
This technique prevents each nuisance estimator from overfitting to the data where it is evaluated, and eschews stringent Donsker conditions on the nuisance estimators, which has been widely used in semiparametric estimation problems <cit.>.




Let K be a fixed positive interger that does not change with sample size N. Take K-fold random partitions {ℐ_k^l}_k = 1^K and {ℐ_k^u}_k = 1^K of the labeled and unlabeled index sets ℐ^l and ℐ^u, respecitvely. Then {ℐ_k = ℐ_k^l ∪ℐ_k^u}_k = 1^K constitutes a K-fold random partition of the whole index set {1, …, N}. 
For each k = 1, …, K, we define ℐ_k^c = {1, …, N}∖ℐ_k, and use all but the kth fold data
to train machine learning estimators for the nuisance parameters:  η̂_k = η̂({W_i}_i ∈ℐ_k^c). The final ATE estimator is δ̂ that solves the following equation: 

    1/K∑_k = 1^K ψ(W_i; δ̂, η̂_k) = 0,

or equivalently,

    δ̂   = 1/K∑_k = 1^K[(1, X) - (0, X) 
                   + TR/(X)(1, X, S)(Y - (1, X, S)) - (1 - T)R/(1 - (X))(0, X, S)(Y - (0, X, S)) 
                    + T/(X)((1, X, S) - (1 , X)) - 1 - T/1 - (X)((0, X, S) - (0 , X))],

where  is the empirical average over the k^th fold, , (1, X) = 1/|_k|∑_i ∈_k(1, X_i).





 §.§ Asymptotic Properties of the Estimator


In this section we establish the insensitivity of our estimator to the estimation of nuisances. Namely, we establish both a double robustness property as well as efficiency. We then proceed to use our results to also construct valid confidence intervals.


Our results will depend on the asymptotic behavior of our nuisance estimates, η̂_k. To state our results, we use the next assumption to define both the limit point of the estimates and the convergence rate. Note it is only an “assumption” once we specify a certain limit point and rate. In particular, the below allows the nuisance estimators to be misspecified in that the limit point η_0 need not be equal to η^*.


[Nuisance Estimator Convergence Rate]
For k = 1, …, K, the nuisance estimators η̂_k = (, , , ) converge to their limit η_0 = (, , , ) in mean sqaured error at the following rates:

    - _2 = O_p(),   - _2 = O_p(),   - _2 = O_p(),   - _2 = O_p().

Furthermore, the propensity score estimators and their asymptotic limits are almost surely bounded: (X), (X) ∈ [ϵ, 1 - ϵ] and (X), (X) ∈ [ϵ, 1] with probability 1. 


We further assume the following boundedness on the variance of the primary outcome.


[Bounded Moments]
There exist constants C > 0, q > 2 such that 

    {Y | X, S, T}_∞≤ C,   {Y | X, T}_∞≤ C, 
       { (T, X, S) | T, X}_∞≤ C,   Y(1)_q ∨Y(0)_q ≤ C



Our next result establishes formally the notion that only one of  and  and one of  and  need be consistent for δ̂ to be consistent.



Given <Ref>, if we further assume that , , , are all o(1), 

    ( - )( - ) = 0,    ( - )( - ) = 0,

and the asymptotic bias of the outcome regressions are almost surely bounded by the positive constant C, , - _∞≤ C and - _∞≤ C, then 

    δ̂p→δ^*  as  N →∞.


<Ref> states that the proposed estimator converges to the true ATE, as long as all nuisance estimators converge to a limit point and at least one of the limit points, but not necessarily both, in each pair of (, ) and (, ) is equal to the corresponding true value. 
Thus the consistency of our estimator does not require all nuisance parameters to be correctly estimated, nor the knowledge of which one is correctly estimated. 
This means that our estimator is robust to misspecification errors of estimating some nuisance parameters, as long as  the rest are consistently estimated. This property is called “double robustness” in causal inference literature <cit.>.

Our next result formalizes the notion that slow convergence rates in nuisance estimation multiply, causing the effect of estimating nuisances to be negligible in analyzing δ̂ and its first-order behavior to be similar to δ̂_0 that uses the true nuisances. 

Under assumptions in <Ref>, if we further assume = o(N^-1/2), 
= o(N^-1/2), and that all nuisance components are correctly specified so that -  =  -  =  -  =  -  = 0, then as N →∞,

    √(N)(δ̂  - δ^*) d→𝒩(0, V^*),

where V^* is the efficiency lower bound in <Ref>.


<Ref> further shows that if all nuisance estimators converge to the truth at sufficiently fast rate, then the proposed estimator δ̂ converges at rate O_p(N^-1/2), and it is asymptotically normal with the efficiency lower bound V^* as its limiting variance. The rate requirement is lax and can be satisfied even if all nuisance estimators converge to true values at o_p(N^-1/4) rates, , much slower than the parametric rate O_p(N^-1/2).
Therefore, the estimation errors of machine learning nuisance estimators may not undermine the asymptotic behavior of our ATE estimator and it can still achieve the efficiency bound of <ref> similarly to the infeasible estimator δ̂_0.

Notably, we do not restrict the nuisance estimators to Donsker or bounded entropy classes <cit.>, which flexible machine learning methods usually violate. 
Instead, we only require a lax rate condition on the nuisance estimators. 
This rate requirement can be achieved by many machine learning estimators in high dimensions under appropriate conditions <cit.>. 
Moreover, we allow estimators converging at faster rate to compensate for those converging at slower rate. 
For example, if we have strong domain knowledge about the labeling process and treatment assignment process (, in a randomized experiment with two-phase sampling design) so that we can estimate the labeling propensity score  and treatment propensity score  at very fast rate, then we can allow for very flexible regression estimators , that converge at slow rates. 

In the next result we propose a way to consistently estimate the efficient variance, V^*, which immediately lends itself to confidence interval construction.


Under the assumptions in <ref>,

    V̂ = 1/K∑_k = 1^K[ψ^2(W; δ̂, η̂_k)] p→ V^*  as  N →∞.

Consequently, the following (1 - α) × 100% confidence interval 

    CI = (δ̂ - Φ^-1(1 - α/2)(V̂/N)^1/2,   δ̂ + Φ^-1(1 - α/2)(V̂/N)^1/2)

with Φ as the cumulative density function of standard normal distribution satisfies that 

    (δ^* ∈CI) → 1 - α as  N →∞.


<Ref> shows that under the same conditions, a consistent estimator for the efficiency lower bound can be obtained by forming the sample analogue of [ψ^2(W; δ^*, η^*)] with cross-fitting nuisance estimators and the proposed estimator δ̂. 
The resulting confidence interval in <ref> asymptotically achieves correct coverage probability. 
Also, since the proposed estimator δ̂ asymptotically achieves the smallest possible variance, the confidence interval in <ref> tends to be shorter than confidence intervals based on other less efficient estimators. 




§ EXTENSION: VERY LARGE UNLABELED DATA

In this section, we consider the setting where the size of unlabeled data is much larger than the size of the labeled data, , N_u ≫ N_l.
This is a practically relevant setting since in many cases we may have only very few units being followed in a study while at the same time we may cheaply collect massive amounts of data without outcomes from existing databases, such as from electronic medical records <cit.>.
Despite its practical relevance, this setting cannot be directly accommodated by our efficiency and estimation theory in <Ref>.
This is because  previous results all hinge on the overlap condition (<ref>) in <Ref>, 
which implies that the 
marginal labeling probability is positive, (R = 1) ≥ϵ, and thus N_u ≍ N_l with high probability. This rules out the setting with many more unlabeled data, that is, N_u ≫ N_l and (R = 1) = 0. 
In fact, in the very-many-unlabeled-data setting, we can show that (T, X, S) = 0 almost surely (<Ref>), so previous efficiency lower bounds based on 1/ are invalid.
Despite these drastic differences, we will show that essentially the same efficiency results and estimation strategy actually still apply in the very-many-unlabeled-data setting, as long as we change the perspective  and scaling appropriately. 



 §.§ Efficiency Analysis


To accommodate this  setting,
we change the efficiency considerations mainly in two aspects. 
First, instead of using , we characterize efficiency in terms of the density ratio 

    λ^*(X) f^*(X)/f^*(X | R  = 1) = f^*(X | R = 1)/f^*(X | R  = 1)(R = 1)_=0 + f^*(X | R = 0)/f^*(X | R  = 1)(R = 0)_=1,

where f^*(·| R = ·) and f^*(· ) are the conditional and unconditional density functions of X corresponding to the target distribution  with respect to an appropriate dominating measure (, Lebesgue measure).
This density ratio λ^* is well-defined and bounded almost surely as long as the distribution of X on the labeled data and that on the unlabeled data sufficiently overlap, regardless of whether N_l ≍ N_u or N_l ≪ N_u. 


Second, we will focus on ATE estimators whose convergence rates scale with the sample size of labeled data instead of the total sample size.  
This is crucial since,
in the current setting, the size of the labeled data becomes the bottleneck for accurate ATE estimation, since the primary outcome observed in the labeled data is the primary source of information on the ATE, but its sample size, N_l, is on a different scale than the total sample size, N.  


Since N_l ≪ N_u, from the perspective of the behavior as N_l→∞, the size of the unlabeled dataset appears infinitely larger and thus its distribution, , the distribution of (X, T, S) given R = 0, appears virtually known. 
Moreover, in the asymptotic limit, the labeled dataset is negligible, and the combined dataset is virtually identical to the unlabeled dataset (N_u/N→1).
Thus the unconditional distribution of (X, T, S) is the same as their conditional distribution  given R = 0 for the unlabeled data,
 thus the unconditional distribution can also be viewed as known from the perspective of labeled data. 
The efficiency lower bound from this perspective is formalized in the following theorem.


Suppose our data consists of i.i.d draws 
from the conditional distribution of (X, T, S, Y) given R = 1.  
Suppose 
<Ref> 
and <Ref> <Ref> hold, the unconditional distribution of (X, T, S) is known, and 
assume an additional simplifying condition
    R ⊥ T | X. Then the efficiency lower bound is Ṽ^* =  [ψ̃^2(W; δ^*, η̃^*) | R = 1], where the new nuisance parameters are η̃^* = (, λ^*, , ), and 

    ψ̃(W; δ^*, η̃^*) = Tλ^*(X)/(X)(Y - (1, X, S)) - (1 - T)λ^*(X)/1 - (X)(Y - (0, X, S)).



<Ref> assumes the MAR assumptions in 
<Ref>,
 which strictly generalizes the results in <cit.> under the more restrictive MCAR condition.
This generalization is possible mainly because we formulate the efficiency lower bound in terms of the density ratio, as opposed to inverse labeling propensity score formulation that is more commonly used but ill-defined in the current setting. 
Although the assumption of exactly knowing the distribution of (X, T, S) in <Ref> seems idealized, we will confirm in <Ref> below that this indeed characterizes the role of unlabeled data in an asymptotic sense from the perspective of labeled data.
In <Ref>, we assume the condition 
R ⊥ T | X to obtain a simplied result. 
In the proof for <Ref>, we also derive the efficiency bound for the more general setting without this condition.



 §.§ Constructing an Efficient Estimator


Our efficiency analysis above is asymptotic, where asymptotically we have N_l≪ N_u and hence (R = 1) = 0. In terms of estimation from an actual finite sample, however, we do have some labeled data, albeit much less than unlabeled data. Therefore, the observation model of having N i.i.d. draws from  is inappropriate as it implies we observe no outcome data with probability 1, which would make estimation impossible. Instead, we modify our observation model slightly to allow for the marginal probability of labeling, which we denote as r_N, to vary with the total sample size N. We only require that r_N>0, r_N→(R=1) and that the expected labeled sample size satisfies _l = r_N N →∞ as N →∞. We then consider the observation model where we have N i.i.d. draws, for each of which with probability r_N we sample the observation from the conditional distribution of (X, T, S, Y) given R=1 and with probability 1-r_N we sample from the conditional distribution given R=0. That is, we define ^(N)(ℰ) = (ℰ| R = 1)r_N + (ℰ| R = 0)1 - r_N for any event ℰ measurable with respect to (X, T, S,Y) and we observe N i.i.d. draws from ^(N). 
This specification aptly models the regime of interest:  even when (R = 1) = 0, 
the labeled data are always available in finite sample, and asymptotically its absolute size grows to infinity despite the fact that its relative size as a fraction of N vanishes as N →∞.
In fact, this specification is very general: it can accommodate both the current setting by letting r_N  → 0, and the original setting (see <Ref>) by letting r_N=(R=1)>0. 

We now follow our estimation strategy in <Ref> to construct an analogous ATE estimator that can attain the efficiency lower bound in <Ref>. 


We take the K-fold random partitions as in <Ref>, and analogously construct nuisance estimators η̂̃̂_k  = (_k, λ̂_k, , ) for k = 1, …, K. The resulting ATE estimator is   

    δ̂^rev   = 1/K∑_k = 1^K_k^N{T/(X)((1, X, S) - (1 , X)) - 1 - T/1 - (X)((0, X, S) - (0 , X)) 
       + (1, X) - (0, X) +  Rλ̂_k(X)/_NT/(X)(Y - (1, X, S)) - Rλ̂_k(X)/_N1 - T/1 - (X)(Y - (0, X, S))},

where _N = N_l/N > 0 estimates r_N, and _k^N is the empirical average operator for data  in the kth fold (in triangular array), , ^N (1, X) = 1/n∑_i ∈_k(1, X_i, N).


This estimator δ̂^rev is almost the same as the previous estimator δ̂ in <Ref>, except that δ̂^rev is based on nuisance estimators for the density ratio λ^*, while δ̂ is based on nuisance estimators for the labeling propensity score .
In <Ref>, we will show that these two estimators are asymptotically equivalent if the overlap condition (<ref>) holds. 
But when the overlap condition indeed holds, we prefer the estimator δ̂ as it only requires estimating the labeling propensity score , , the conditional probability function in a classification task with the label R, for which many machine learning methods exist and which is arguably easier to estimate than the density ratio λ^* in δ̂^rev. 
When the overlap condition does not hold and N_l ≪ N_u, estimating the labeling propensity score may be very unstable given the extremely imbalanced data, with imbalance going to an extreme as N→∞. Therefore, in the very-large-unlabeled-data setting, we prefer the estimator δ̂^rev as it uses the density ratio λ^*, which is stable as N varies. Machine learning estimators for density ratios are reviewed in <cit.>.  


In the following theorem, we show the asymptotic normality and efficiency of estimator δ̂^rev under conditions analogous to those in <Ref>. Notably, we consider a generic limiting marginal probability (R = 1) that can be either 0 (, the setting in this section) or strictly positive (, the setting in <Ref>).
 

Suppose that the assumptions in <Ref> hold, r_N > 0 for any finite N, r_N →(R = 1) and = r_N N →∞ as N →∞, and mild moment regularity conditions given in Appendix <Ref>. We assume that for 1 ≤ k ≤ K, the convergence rates of nuisance estimators , λ̂_k, , to the true nuisance parameters are O_p(ρ_N, e), O_p(ρ__l, λ), O_p(ρ__l, μ),  and O_p(ρ__l, μ̃) respectively (see Appendix <Ref> for precise definitions), λ̂_k(X)_∞≤ C, and (X) ∈ [ε, 1- ε] almost surely. We further assume that , ρ__l, μ, ρ__l, λ, ρ__l, μ̃ are all o(1) and r_N/r̂_N - 1 = o_p(1). Moreover, we suppose that they satisfy ρ__l, λρ__l, μ̃ = o(_l^-1/2), 

ρ_N, eρ__l, μ =  o(_l^-1/2),
and r_N/r̂_N - 1ρ__l,  = o_p(_l^-1/2). Then estimator δ̂^rev given in <Ref> satisfies that 

    ^1/2_l(δ̂^rev - δ^*) d→𝒩(0, Ṽ),  as   N →∞, 
    where Ṽ   = [ψ̃^2(W; δ^*, η̃^*) | R = 1] + (R = 1)[(1, X) - (0, X) - δ^*]^2   
       +(R = 1)[T/(X)((1, X, S) - (1 , X)) - 1 - T/1 - (X)((0, X, S) - (0 , X))]^2.

If the overlap condition in <ref> holds so that (R = 1) > 0, and we further assume the conditions in <Ref>, then estimator δ̂ in <Ref> is asymptotically equivalent to estimator δ̂^rev, , 

    ^1/2_l(δ̂ - δ^*) = ^1/2_l(δ̂^rev - δ^*) + o_p(1) d→𝒩(0, Ṽ),  as   N →∞.

If (R = 1) = 0, then Ṽ is equal to the efficiency lower bound Ṽ^* given in <Ref>. 


<Ref> shows that if the  overlap condition in <ref> holds and nuisance estimators are accurate enough, then estimators δ̂^rev and δ̂ are asymptotically equivalent, and they are both asymptotically efficient. 
Their asymptotic variance Ṽ can be viewed as an equivalent reformulation of the efficient lower bound V^* in <Ref> from the perspective of the labeled data.
In particular, in contrast to the scaling factor of N^1/2 and the inverse labeling propensity score 1/ in V^*, Ṽ is formulated in terms of the scaling factor ^1/2_l and the density ratio λ^* appearing in ψ̃(W; δ^*, η̃^*).
This new formulation is not only valid under the overlap condition, but also particularly suitable for the current setting where (R = 1) = 0. 
In the current setting, although the estimated  labeling probability _N in δ̂^rev 

converges to 0, the scaling factor ^1/2_l effectively stabilizes the asymptotic variance of δ̂^rev and ensures meaningful asymptotic distribution (see <ref>). In contrast, the scaling factor N^1/2 will be far too large: if the N data points consist mostly of unlabeled data, we cannot hope for O_p(N^-1/2)-rate convergence. 
Moreover, the density ratio formulation circumvents the inverse labeling propensity score 1/ that is ill-defined in the current setting. 

<Ref> also demonstrates that our efficiency analysis in <Ref> is appropriate: in the regime where N_l / N_u→0, our estimator achieves the efficiency bound in <ref> where the unconditional distribution of (T, X, S) is assumed to be known.
In particular, the last two terms in the asymptotic variance Ṽ, which  involve the limiting labeling probability (R = 1),  quantify the estimation variance due to not knowing the unconditional distribution of (T, X, S). When (R=1)=0, 
this unconditional distribution can be viewed as known from the perspective of labelled data, so 
the last two variance terms drop out and only the first term remains. 
These efficiency and estimation results thus reveal the whole spectrum of efficiency in estimating ATE with surrogates, and feature a smooth transition from the regime N_l ≍ N_u to the regime N_l ≪ N_u.
In Appendix <Ref>, we further extend <Ref> to the regime N_l ≪ N_u, and we again confirm that more predictive surrogates result in bigger efficiency gains.


All results above depend on the density ratio λ^*, which might be challenging to estimate in practice, especially if the pre-treatment covariates are high dimensional. One straightforward setting where we can avoid estimation λ^* is when the primary outcome is MCAR, which was studied in <cit.>.  In this special setting, R is independent with all other variables, so λ^*(X) = f^*(X)/f^*(X | R = 1) = 1 does not need to be estimated, and the estimator δ̂^rev in <Ref> reduces to

    δ̂^rev   =1/K∑_k = 1^K_k{T/(X)((1, X, S) - (1 , X)) - 1 - T/1 - (X)((0, X, S) - (0 , X))
       +(1, X) - (0, X) +  TR/(X)r̂_N(Y - (1, X, S)) - (1 - T)R/(1 - (X))r̂_N(Y - (0, X, S))}.

This estimator  is exactly identical to the estimator δ̂ in <Ref> under the MCAR assumption. 
In <Ref>, we further discuss how this estimator connects to those in <cit.>, and how our result generalizes those in previous literature. 




§ NUMERICAL STUDIES

In this section, we demonstrate the performance of our proposed estimators using experimental data for the Greater
Avenues to Independence (GAIN) job training program, a  job assistance
program designed in the late 1980s for low-income people  in California.
We employ the dataset analyzed in <cit.>, which  contains results from a large-scale randomized experiment in four counties in California (Alameda, Los Angeles, and San Diego and Riverside). 
For
each experiment participant, this dataset records a binary treatment variable indicating whether
being treated by the GAIN program or not, quarterly earnings after treatment
assignments, and other covariate information, etc.
In this section, we illustrate the performance of our proposed estimator and other benchmarks in estimating the average treatment effect in long-term earnings using the Riverside data. 
We provide additional results for Los Angeles data and San Diego data in <Ref> (Alameda dataset is very small and thus omitted).
The code underlying this numerical study is available at <https://github.com/CausalML/Efficient_estimation_surrogate>.

We construct a labelled dataset and unlabelled dataset based on 
the Riverside data (N = 5445, with 4405 treated units and 1040 control units). 
We randomly draw a fraction  r ∈0.1, 0.3, 0.5 of units from the Riverside data as the labelled data (R=1) and use the rest as the unlabelled data (R=0). 
In our analysis, the primary outcome is the long-term earning in the 36th quarter after the treatment assignment, and  surrogates are quarterly earnings up to s quarters after the treatment, where s ∈8, 16, 24, 32.
We also consider additional covariates including age, gender, education, and ethnicity. 
The surrogates and covariates are observed on both labelled and unlabelled data, but the primary outcome is only observed on the labelled data. 





We apply five types of estimators to  
estimate the average treatment effect in the primary outcome: our proposed estimator in <Ref> (denoted as “Proposal”), the surrogate index estimator based on regression imputation proposed in <cit.> (denoted as “SIndex_REG”), the semiparametrically efficient surrogate index estimator proposed in <cit.> (denoted as “SIndex_DR”), the doubly robust estimator that uses both datasets but ignores the surrogates, corresponding to the influence function ψ_I in <Ref>  (denoted as “DR” with type s = 0), and the doubly robust estimator that only uses the labelled dataset (denoted as “DR” with type “Labelled”). 
We estimate the nuisance functions in these estimators by fitting random forests, gradient boosting, and LASSO respectively, all cross-fitted with K = 5 folds. 
The SIndex_DR estimator is implemented by the longterm R package developed by <cit.>, where all hyperparameters in nuisance estimation are automatically tuned by cross-validation. 
All other estimators (including our proposal) are implemented using R packages ranger (for fitting random forests), gbm (for fitting gradient boosting), and glmnet (for fitting  LASSO). The hyperparameters in random forests and boosting are set as default values without any tuning, and those in LASSO are tuned by cross-validation.
Note that in our data generating process, both the treatment assignments and the primary outcome missingness are completely at random, so we directly use sample frequencies as estimates for the corresponding propensity nuisances.
These propensity score estimates are trivially consistent. 
In this section, we focus on estimation results with random forest nuisance estimators. 
Additional results for other nuisance estimators are provided in the appendix. 


The upper row in <Ref> shows the bias  of different treatment effect estimators with nuisances estimated by random forests, over 120 repetitions of the experiments on the Riverside data.  
We observe that the surrogate index  estimators always have high bias. 
This bias is not primarily caused by nuisance estimation, since according to the theory in <cit.>, the efficient surrogate index estimator (SIndex_DR) should be robust to the nuisance estimation bias given that the propensities are well estimated (its bias is indeed lower than the bias of SIndex_Reg). 
Instead, the high bias may be plausibly attributed to  the violation of the statistical surrogate condition assumed for the surrogate index methods. 
As the number of surrogates grows, the violation of the statistical surrogate condition is alleviated, so the bias of surrogate index estimators drops, but it 
still remains very high, posing serious challenges for statistical inference. 
In contrast, our proposed estimator has very low bias across all settings.

The lower row in <Ref> shows the standard errors of different treatment effect estimators.
We note that the standard errors of our proposed estimator decrease with the number of surrogates.
This is because with more surrogates we can better predict the primary outcome, as we validate in <Ref>.
Moreover, the amount of standard error reduction due to introducing surrogates is overall higher when the missingness of the primary outcome is more severe (smaller r). 
These empirical observations support our qualitative conclusions from the efficiency analysis in <Ref>: the efficiency gains from leveraging surrogates improve for more predictive  surrogates and more missing primary outcome. 
Interestingly, we find that the standard errors of surrogate index estimators may not decrease with the number of surrogates. 

In <ref>, we provide additional results for other types of nuisance estimators (Gradient Boosting and LASSO) and  results for Los Angeles data and San Diego data.
These additional results show similar patterns of bias and standard error.  



§ CONCLUSION

We study the estimation of average treatment effect with only a limited number of primary outcome observations but  abundant observations of surrogates. 
Particularly, we avoid stringent surrogate conditions that are prone to violation in practice and only assume standard causal inference and missing data assumptions. 

We investigated the role of surrogates by comparing the efficiency lower  bounds of ATE  with and without presence of surrogates, and also bounds in some intermediary cases. 
We find that efficiency gains from optimally leveraging surrogates crucially depend on how well surrogates can predict the primary outcome and also the fraction of missing outcome data.
These results provide valuable insights on when leveraging surrogates can be beneficial. 
We also show that the efficiency results are valid in two regimes: when the size of surrogate observations is comparable to the size of primary-outcome observations   (, N_u ≍ N_l), and when the former is much larger than the other (, N_u ≫ N_l).
The second regime violates the overlap condition commonly assumed in the literature and was thus understudied in the past, even though it is highly relevant in modern data collection. 
Our analysis shows that the second regime can be viewed as a limiting case of the first regime, which reveals the intimate connection between these two regimes. 

Moreover, we propose ATE estimators that can employ any flexible machine learning method for nuisance parameter estimation. We provide strong statistical guarantee for the proposed estimators by showing that they are robust to nuisance estimation bias, and they asymptotically achieve the semiparametric efficiency lower bounds under high-level rate conditions for the machine learning nuisance estimators. 
We further develop consistent estimators for the efficiency lower bounds and construct asymptotically valid confidence intervals for ATE. 
In summary, our methods provide a principled approach to optimally leverage surrogate observations when only a limited number of primary-outcome observations are available and without using strong surrogacy assumptions. 

plainnat





Appendices





§ STATISTICAL SURROGATE CONDITION

In this section, we review the definition of statistical surrogate condition proposed by <cit.>. Throughout this section, we implicitly condition on pre-treatment variables X in all distributional statements. For example, Y ⊥ T | S stands for Y ⊥ T | S, X.

 <cit.> suggested a valid surrogate S satisfy that a test of the null of no effect of the treatment T on surrogate S should serve as a valid test of the null of no effect of treatment T on outcome Y. They formalized this by the following “statistical surrogate” condition. 


S is said to
be a surrogate for the effect of T on Y if (i) Y ⊥ T | S; (ii) S and Y are correlated.


To justify this condition, <cit.> considered a time-to-event primary outcome with surrogates sampled from a stochastic process.
For simplicity, we now adapt their argument to a single-time measurement case. 
Note that under the statistical surrogate condition, we can easily show that  

    F(y | t) = ∫ F(y | t, s)dF(s | t) =  ∫ F(y | s)dF(s | t),

where F(y | t), F(y | t, s), F(s | t) are conditional cumulative distribution functions for the corresponding random variables. This equation shows that under the statistical surrogate condition, T is dependent with Y only if T is dependent with S. 
See also <cit.> for a similar argument for binary outcome. 
However, this type of argument is based purely on the statistical relationship rather causal relationship among the treatment, surrogate, and the primary outcome. 
Thus, the causal implication of this argument is not immediately straightforward. 

In the language of causal diagram <cit.>, the statistical surrogate condition is often characterized by <Ref> <cit.>. In this diagram, T has no direct effect on Y, and S has an effect on Y. As a result, T can have an effect on Y only if T has an effect on S. 
Also, no direct effect of T on Y implies that T is independent of Y given S, namely the condition (i) in the definition of statistical surrogate. 
However, this relationship may be invalidated by any unmeasured confounder between the surrogate and the primary outcome (, the variable U in <Ref>):
since S is a collider on the causal path T → S ← U → Y, conditioning on S can induce spurious dependence between T and Y,  even though there is no direct effect of T on Y <cit.>.
In other words, no direct effect of the treatment T on the primary outcome Y does not necessarily ensure conditional independence between the treatment T and primary outcome Y given surrogates S, if there exists any unmeasured confounder between surrogates S and the primary outcome Y.

The following proposition, adapted from Proposition 3 in <cit.>, reiterates the implication of <Ref> in language of potential outcomes, and further elucidates the causal assumptions underlying the statistical surrogate condtion. We denote Y(t, s) as the potential outcome that would have been realized if treatment T had been set to t, and surrogate oucomes S had been set to s.


S satisfies condition (i) in <Ref> if the following conditions hold:


  * Y(t, s) = Y(t', s) for any t, t' ∈{0, 1} and s ∈𝒮; ;

  * T ⊥ (Y(0, s), Y(1, s))_s ∈𝒮;  

  * S(t) ⊥{Y(t, s)}_s ∈𝒮| T = t for any t ∈{0, 1}. 



<Ref> above shows that no direct effect of treatment on the primary outcome (condition <ref>), and no unmeasured confounding either between treatment and the primary outcome (condition <ref>) or between surrogates and primary outcome (condition <ref>) together ensure statistical surrogate condition. 
Conditions <ref><ref> are also commonly assumed in mediation analysis that aims to decompose the total effect of treatment T into the direct effect not through post-treatment variable S and the effect mediated by S <cit.>.
Here condition <ref> may be satisfied by design in randomized trials where the treatment assignment T is under perfect control. 
However, surrogates S and their relationship to the primary outcome are generally  not manipulatable, so <ref> and <ref> are often (if not always) violated even in perfect randomized trials.  



The discussions above also reveal that it is perhaps misleading to follow the quite common practice of interpreting statistical surrogates as variables that block all causal pathways between the treatment and primary outcome (, no-direct-effect assumption characterized by condition <ref> in <Ref>). Actually, the no-direct-effect condition is neither sufficient nor necessary for the conditional independence between the treatment and the primary outcome given surrogates (, condition (i) in <Ref>), since there may exist unmeasured confounders between the surrogates and the primary outcome (, condition <ref> in <Ref> is violated).  
For example, section 5.2 in <cit.> provide counter-examples 
to show that statistical surrogates may not satisfy no-direct-effect condition and vice versa. 



§ COMPARISONS WITH PREVIOUS LITERATURE






 §.§ Comparison with <cit.>

<cit.> consider the same data configuration as our paper (<Ref>), but they assume that the primary outcome is missing completely at random. 

Recall that our estimator under MCAR setting reduces to 

    δ̂^rev   =1/K∑_k = 1^K{T/(X)((1, X, S) - (1 , X)) - 1 - T/1 - (X)((0, X, S) - (0 , X))
       + (1, X) - (0, X) + TR/(X)_N(Y - (1, X, S)) - (1 - T)R/(1 - (X))_N(Y - (0, X, S))}.


This  estimator and the estimator in <cit.> both asymptotically achieve the efficiency lower bound  in <Ref> with λ^*(X) = 1. 
The estimator in <cit.> is valid only under MCAR setting,
while our estimator can be straightforwardly extended to MAR setting, if augmented with a density ratio estimator (<Ref>).
Moreover, the estimator in <cit.> imposes parametric assumptions on the nuisances and relies on computationally intensive resampling methods to construct confidence intervals.  
In contrast, our estimator can leverage the power of any flexible machine learning nuisance estimator under generic rate conditions, and its confidence interval can be easily constructed using a straightforward plug-in estimator for standard errors (<Ref>).
Furthermore, <cit.> focuses on the setting of N_l ≪ N_u, while our analysis accommodates both N_l ≪ N_u and N_l ≍ N_u, and reveals the the whole spectrum of efficiency limits across two regimes. 



 §.§ Comparison with <cit.>
 
<cit.> focus on the efficiency improvement from unlabeled data, without studying possible efficiency gains from incorporating surrogates (<Ref>, or equivalently the setting I in <Ref>).

This setting can be viewed as a special case of our problem: we can view S as an empty set of random variables and thus (T, X, s) = (T, X) for any s ∈𝒮. Consequently, our estimator in <ref> corresponding MCAR primary outcome reduces to the following form:  

    1/N∑_i = 1^N [μ̂_k(i)(1, X_i) - μ̂_k(i)(0, X_i)] 
               + 1/N_l∑_i ∈ℐ_l[T_i/ê_k(i)(X_i)(Y_i - μ̂_k(i)(1, X_i)) - 1 - T_i/(1 - ê_k(i)(X_i))(Y_i - μ̂_k(i)(0, X_i))],

where k(i) is the fold that the ith observation belongs to. 
This estimator  recovers the semi-supervised ATE estimator in <cit.>. 




 §.§ Comparison with <cit.>

In <cit.>, they assumed the statistical surrogate condition that Y ⊥ T | X, S, R = 1, namely the observed primary outcome and the treatment on the labeled data are independent given the pre-treatment covariates and surrogates.
This assumption is crucial for the identification of treatment effects in the  setting considered by <cit.>: the treatment and primary outcome are observed on separate datasets, but surrogates are always observed (<Ref>). 
Their setting is different and more challenging than our setting: in our setting the treatment is always observed (<Ref>), but in their setting the treatment is missing on the labelled data.
Although the statistical surrogate condition seems inevitable in their setting to fuse the two separate datasets without any complete observation, the causal assumptions underlying this statistical surrogate condition may be too strong to hold in practice, as we discussed in  <Ref>.





§ SUPPLEMENTS FOR <REF>



 §.§ Assumptions

In this part, we give some supplementary assumptions for <Ref>.


There exist positive constants q > 2 and C such that for t = 0, 1,

    {[|Y - (T, X, S)|^q| R = 1]}^1/q≤ C, 
       (t, X, S) - (t , X)_q ≤ C,    (t , X)_q  ≤ C.


Moment conditions in <Ref> are mainly used in verifying the Lyapunov condition in Lindberg-Feller Central Limit Theorem in the proof of <Ref>.



Suppose that the nuisance estimators have the following convergence rates: for 1 ≤ k ≤ K, and r = 0, 1,

    {[((X) - (X))^2 | R = r]}^1/2   = O_p(ρ_N, e),    {[(λ̂_k(X) - λ^*(X))^2 | R = r]}^1/2 
        = O_p(ρ__l, λ), 
    {[((X) - (X))^2 | R = r]}^1/2   = O_p(ρ__l, μ),    {[((X) - (X))^2 | R = r]}^1/2 
        = O_p(ρ__l, μ̃).



In <Ref>, we specify the convergence rates of the nuisance estimators in mean squared error both on the labeled data and on the unlabeled data.  
Since  is trained on the whole data where the treatment variable is always fully observed, its convergence rate  scales with the total sample size N. In contrast, the convergence rates of  and , , ρ__l, μ and ρ__l, μ̃, scale with _l because they are trained only on the labeled data. 
The convergence rate of λ̂_k is also constrained by the expected size of labeled data _l, because it involves estimating the density of covariates on the labeled data, , f^*(X | R = 1).




 §.§ Efficiency Comparison

We now provide the efficiency lower bounds for other settings in <Ref> when N_l ≪ N_u. Note that setting IV is the ideal setting with fully labeled data, so in the regime of N_l ≪ N_u, the distribution of all data can be viewed as known and the efficiency bound degenerates to 0.
Therefore, we only need to study setting I and setting II. 


Consider the following two settings:


  * We only observe the labeled data, , i.i.d. samples from the conditional distribution of (X, T, Y) given R = 1, and we know the unconditional distribution of (X, T);

  * We only observe the labeled data, , i.i.d. samples from the conditional distribution of (X, T, S, Y) given R = 1, and we know the unconditional distribution of (X, T);

We further assume assumptions in <Ref>.  
Then the efficiency lower bounds for two settings above are Ṽ^*_j = [ψ̃^2_j(W; δ^*, η̃^*) | R = 1] for j = I, II, where 

    ψ̃_I(W; δ^*, η̃^*) = ψ̃_II(W; δ^*, η̃^*)  = Tλ^*(X)/(X)(Y - (1, X)) - (1 - T)λ^*(X)/1 - (X)(Y - (0, X)).

The efficiency gains from surrogates are quantified by 

    Ṽ^*_I - Ṽ^*    = Ṽ^*_II - Ṽ^*
       = [λ^*2(X)/(X){(1, X, S(1))| X} + λ^*2(X)/1 - (X){(0, X, S(0))| X}| R = 1].

 

<Ref> shows that the efficiency gains from surrogates increase with the variations of the primary outcome explained by the surrogates beyond the pre-treatment covariates, , {(t, X, S(t))| X} for t = 0, 1. This means that surrogates that are more predictive of the primary outcome can result in larger efficiency improvement, which is in line with the findings in <Ref>.







 §.§ MCAR Setting

To further illustrate the efficiency improvement from the unlabeled data, we compare the efficiency lower bounds of ATE under three different settings: only labeled data are available (Setting i), both labeled data and unlabeled data are available and their sizes are comparable (Setting ii), and size of the unlabeled data is much larger than size of the labeled data (Setting iii).
We assume that the primary outcome is MCAR, , R ⊥ (X, T, S(1), S(0), Y(1), Y(0)), so that we can estimate ATE based on labeled data alone.


 
  Setting i: only labeled data. We observe M i.i.d. samples from the distribution of (X, T, S, Y) given R = 1. 
 
  Setting ii: labeled data and unlabeled data of comparable size. We observe N i.i.d. samples from the distribution of (X, T, S, Y, R) with Y = NA when R = 0. The total sample size N satisfies that N(R = 1) = M, , the expected size of labeled data is also M.
 
  Setting iii: much more unlabeled data than labeled data. We observe M i.i.d. samples from the distribution of (X, T, S, Y) given R = 1, and we additionally know the marginal distribution of (X, T, S). According to <Ref>, this setting also asymptotically characterizes the regime where we have access to unlabeled data whose size is much larger than the  size of the labeled data, but we do not know the distribution of (X, T, S) exactly (see <Ref>).
 
We now provide the efficiency lower bounds in these three settings to illustrate efficiency improvement from increasingly more unlabeled data information. Notably, we follow <Ref> to formulate the efficiency lower bounds from the perspective of the labeled data (see <Ref> for more interpretations). 


Suppose <Ref> and <Ref> condition (<ref>) hold. 
Further assume that  R ⊥ (X, T, S(1), S(0), Y(1), Y(0)) hold.
Then for j = i, ii, iii, the efficiency lower bound for ATE from the perspective of the labeled data (see <Ref> for precise interpretations) in setting j is given by V^*_j, where 

    V^*_i  
           = [T/(X)(Y - (1, X)) - (1 - T)/1 - (X)(Y - (0, X))]^2 + [(1, X) - (0, X) - δ^*]^2, 
    
    V^*_ii   = [T/(X)(Y - (1, X, S)) - (1 - T)/1 - (X)(Y - (0, X, S))]^2 
       + (R = 1)[(1, X) - (0, X) - δ^*]^2 
       + (R  = 1)[T/(X)((1, X, S) - (1 , X)) - 1 - T/1 - (X)((0, X, S) - (0 , X))]^2, 
    
    V^*_iii   =  [T/(X)(Y - (1, X, S)) - (1 - T)/1 - (X)(Y - (0, X, S))]^2.

The efficiency gain from observing the unlabeled data in setting ii compared to setting i is 

    V^*_i - V^*_ii   = (1 - (R = 1))[1/(X){(1, X, S(1))| X} + 1/1 - (X){(0, X, S(0))| X}] 
       + (1 - (R = 1))[(1, X) - (0, X) - δ^*]^2.

The further efficiency gain from knowing the marginal distribution of (X, T, S) in setting iii (or equivalently observing unlabeled data whose size is much larger than M) is 

    V^*_ii - V^*_iii   = (R = 1)[1/(X){(1, X, S(1))| X} + 1/1 - (X){(0, X, S(0))| X}]_V_S
       + (R = 1)[(1, X) - (0, X) - δ^*]^2_V_X.



<Ref> shows that efficiency improvement from optimally leveraging extra unlabeled data depends on  the relative size of the unlabeled data (characterized by 1 - (R = 1)), and variations of surrogates S and pre-treatment covariates X (characterized by V_S and V_X respectively). In the limit, if the size of the unlabeled data is much larger than the size of the labeled data, so that from the perspective of labeled data we virtually know the unlabeled data distribution, then only the residual variations of Y not explained by X, S, , V^*_iii remains in the efficiency lower bound. 
This analysis supplements the efficiency comparisons in <Ref>.






§ PROOFS





 §.§ Supporting Lemmas


Under <Ref>, the following holds:

    [(T, X, S) | T, X] = (T, X).



Under <Ref>, we have that for t = 0, 1,

    [(T, X, S) | T = t, X]
           = [[Y(t) | R = 1, T = t, X, S(t)] | T = t, X] 
       = [[Y(t) | T = t, X, S(t)] | T = t, X] 
       = [Y(t) | T = t, X].

Moreover, under <Ref>, we have (Y(t), S(t)) ⊥ R | T, X. 
This in turn implies Y(t) ⊥ R | T, X. It follows that 

    (t, X) = [Y(t) | T = t, X, R = 1] = [Y(t) | T = t, X] = [(T, X, S) | T = t, X].







If (R = 1) = 0, then (T, X, S) = (R = 1 | T, X, S) = 0 almost surely. 


Obviously [(T, X, S)] = (R = 1) = 0. 

Denote 𝒜 = {(R = 1 | T, X, S) > 0} and 𝒜_m = {(R = 1 | T, X, S) ≥1/m} for m = 1, 2, …. Oviously 𝒜 = ∪_m = 1^∞𝒜_m. By Chebyshev inequality,  

    0 ≤(𝒜_m) ≤ m [(T, X, S)] = 0.

This implies that (𝒜_m) = 0. By the countable subadditivity of probability measure, we thus have (𝒜) ≤∑_m = 1^∞(𝒜_m) = 0. 




For k = 1, …, K, if - _2 = O_p(),   - _2 = O_p(), and - ^2 = O_p(), then for t = 0, 1, 

    (t, X) - (t, X)_2 = O_p(), 
       (t, X, S(t)) - (t, X, S(t))_2 = O_p(), (t, X, S(t)) - (t, X, S(t))_2 = O_p().

If - _2,  - _2 are almost surely bounded, then (t, X, S(t)) - (t, X, S(t)) _2, (t, X) - (t, X)_2 for t= 0, 1 are also almost surely bounded. 

Moreover, if Y(0)_q ∨Y(1)_q  ≤ C for a constant q > 2, then (t, X)_q ≤ C, (t, X, S)_q ≤ C for t = 0, 1. 


We note that 

    - _2 
           = {[(T, X) - (T, X)]^2}^1/2
       = {[((1, X) - (1, X))^2(X) + ((0, X) - (0, X))^2( 1- (X))]}^1/2
       ≥ (2ϵ)^1/2[(1, X) - (1, X)_2 ∨(0, X) - (0, X)_2].

Thus - _2 = O_p() implies (t, X) - (t, X)_2 = O_p() for t = 0, 1. Similarly, we can prove that (t, X, S(t)) - (t, X, S(t))_2 = O_p() given - _2 = O_p(), and (t, X, S(t)) - (t, X, S(t)) _2, (t, X) - (t, X)_2  are almost surely bounded given that - _2,  - _2 are almost surely bounded. 

Moreover, 

    - ^2_2 
           = [((T, X, S) - r(T, X, S))^2] 
       = [[[((T, X, S) - r(T, X, S))^2 | X, T] | X]] 
       = [(X)[((1, X, S(1)) - r(1, X, S(1)))^2 | X, T = 1] 
       + ( 1- (X))[((0, X, S(0)) - r(0, X, S(0)))^2 | X, T = 0] ]  
       =  [(X)((1, X, S(1)) - r(1, X, S(1)))^2  + ( 1- (X))((0, X, S(0)) - r(0, X, S(0)))^2]  
       ≥ 2ϵ((1, X, S(1)) - (1, X, S(1))_2^2  ∨(0, X, S(0)) - (0, X, S(0))_2^2 )

Therefore, (1, X, S(1)) - (1, X, S(1))_2 = O_p() and (0, X, S(0)) - (0, X, S(0))_2 = O_p().

For the last statement, note that 

    (1, X)_q = [^q[Y(1) | X]]^1/qJensen's inequality≤Y(1)_q ≤ C.

 Similarly we can prove that (0, X)_q, (0, X, S)_q ≤Y(0)_q and  (1, X, S)_q ≤Y(1)_q. Thus ψ(W; δ^*, η^*)_q  = O(1).

 



 §.§ Proofs for Section 1


Note that we have 

    [Y(t)]  
           = {[Y(t) | X]}
       = {[(Y(t) | X, S(t)) | X]}
       = {[(Y(t) | T = t, X, S(t)) |  X]}<Ref>
       = {[(Y(t) | T = t, X, S(t), R = 1) |  X]}<Ref>
       = {[(Y(t) | T = t, X, S(t), R = 1) |  X, T = t]}.  <Ref>


It follows that  

    [Y(t)]   
           = {[(Y(t) | T = t, X, S(t), R = 1) |  X, T = t]}
       = {[(Y | T = t, X, S, R = 1) |  X, T = t]}.





 §.§ Proofs for <ref>




The identification of ATE in setting III is already established in <Ref>, so we focus on establishing identification in the other three settings. 

Under <Ref>, we have that (Y(t), S(t)) ⊥ (T, R) | X. This in particular implies that Y(t) ⊥ (T, R) | X. 
Therefore, we have 

    [Y(t)] 
           = [(Y(t) | X)] 
       =  [(Y(t) | T = t, R = 1, X)] 
       = [(Y | T = t, R = 1, X)].

The last display only depends on distributions of observed data in setting I, II, IV in <Ref>.
This shows the identification of ATE in these three settings.



If <Ref> hold, then we have (Y(t), S(t)) ⊥ (T, R) | X. This obviously implies 

    (Y(t), S(t)) ⊥ T | X, R = i,    i ∈0, 1.

Moreover, when <Ref> holds, we have S(t) ⊥ R | T = t, X, 
which is equivalent to S ⊥ R | T = t, X. 
Thus we have (t, x, s) = R = 1 | T = t, X,  S = R = 1 | T = t, X.
Plus, 

    (t, x) 
          = [(T, X, S) | T = t, X = x] 
       = [(t, X, S(t)) | T = t, X = x] 
       = [(t, X, S(t)) | T = t, X = x, R = 1] 
       = [(t, X, S) | T = t, X = x, R = 1] 
       = [Y | T = t, X, S, R = 1| T = t, X = x, R = 1] = Y | T = t, X, R = 1.




Suppose that distribution of X, conditional distributin of S | X, T, and conditional distribution of Y | R, S, T, X have true density functions f^*_X, f^*_S | X, T, f^*_Y | R, S, T, X with respect to a certain dominating measure. We consider the following  model:

    ℳ_np   = {f_X, T, S, R, Y(X, T, S, R, Y) = f_X(X)[e(X)^T(1 - e(X))^1 - T]f_S | X, T(S | X, T) 
                                 × [r(T, X, S)^R(1 - r(T, X, S))^1- R]f^R_Y | R = 1, S, T, X(Y, S, T, X):  
                f_X, f_S | X, T, f_Y | R = 1, S, T, X are arbitrary density functions of the distributions
               indicated by their respective subscripts, and  e(X), r(T, X, S)  are arbitrary
               functions obeying e(X) ∈ [ϵ, 1- ϵ], r(T, X, S) ∈ [ϵ, 1]}
 
The tangent space corresponding to this model is 

    Λ = Λ_X ⊕Λ_T | X⊕Λ_S | T, X⊕Λ_R | S, T, X⊕Λ_Y | R, S, T, X,

where Λ_X, Λ_T | X, Λ_S | T, X, Λ_R | S, T, X, Λ_Y | R, S, T, X are mean square closures of the following sets respectively:

    Λ_X 
           = {_X(X) ∈ L_2(X): [_X(X)] = 0}, 
    Λ_T | X   = {_T | X(T, X) ∈ L_2(T, X): [_T | X(T, X) | X] = 0}, 
    Λ_S | X, T   = {_S| X, T(S, X, T) ∈ L_2(S, X, T): [_S | X, T(S, X, T) | X, T] = 0}, 
    Λ_R | S, X, T   = {_R| S, X, T(R, S, X, T) ∈ L_2(R, S, X, T): [_R | S, X, T(R, S, X, T) | S, X, T] = 0}, 
    Λ_Y | R, S, X, T   = {R ×_Y | R = 1, S, X, T(Y, R, S, X, T) ∈ L_2(Y, R, S, X, T): 
                       [_Y | R = 1, S, X, T(Y, R, S, X, T) | R = 1, S, X, T] = 0}.
 

We now derive the efficient influence function of ξ_1^* = Y(1). The efficient influence function of  ξ_0^* = Y(0) is analogous so we omit the details for brevity. 

According to <Ref>, 

    ξ_1^* = [Y(1)] = [[[Y | R = 1, T = 1, X, S]| X, T = 1]].

Consider regular parametric submodels indexed by parameters γ, where γ = 0 corresponds to the underlying true data distribution. 
We use _γ to denote the expectation under the submodel distribution with parameter value γ.
 Then the corresponding target parameter is 

    _γ[Y(1)] = _γ[_γ[_γ[Y | R = 1, T = 1, X, S]| X, T = 1]].


We also use (Y, R, S, T, X) to denote the score function corresponding to the parametric submodel.
According to the discussions above, we can write 

    (Y, R, S, T, X) = ( X) + (T | X) + (S | T, X)
    + (R | S, T, X) + 
    (Y | R, S, T, X),

where the components above satisfy the restrictions imposed in the sets  Λ_X, Λ_T | X, Λ_S | T, X, Λ_R | S, T, X, Λ_Y | R, S, T, X respectively.

We will next show that 

    ∂/∂_γ[Y(1)]|_γ = 0 = ψ_1(Y, R, S, T, X)(Y, R, S, T, X),

where 

    ψ_1(Y, R, S, T, X) =    (1, X) - ξ_1^* + T/(X)((1, X, S) - (1 , X)) 
       + TR/(X)(1, X, S)(Y - (1, X, S))).


Note that 

    ∂_γ[Y(1)]/∂γ|_γ = 0   = ∂/∂γ_γ[(1, X)]|_γ = 0 + [∂/∂γ_γ[(1, X, S) | X, T = 1] |_γ = 0] 
       + [[∂/∂γ_γ[Y | T = 1, R = 1, X, S]|_γ = 0| X, T = 1]] .

Now we deal with each term respectively.

First, 

    ∂/∂γ_γ[(1, X)]|_γ = 0   = (1, X)(X) = (1, X) - ξ_1^*(X)
       = (1, X) - ξ_1^*(Y, R, S, T, X).


Second, 

    [∂/∂γ_γ[(1, X, S) | X, T = 1] |_γ = 0] 
          = (1, X, S)(S | X, T) | X, T = 1
       = (1, X, S) - (1, X)(S | X, T) | X, T = 1
       = (1, X, S) - (1, X)(Y, R, S, T, X) | X, T = 1
       = T/e^*(X)(1, X, S) - (1, X)(Y, R, S, T, X).


Third, 

    [[∂/∂γ_γ[Y | T = 1, R = 1, X, S]|_γ = 0| X, T = 1]] 
    
    =   [[[Y×(Y | R, S, T, X) | T = 1, R = 1, X, S]] | X, T = 1]] 
    
    =   [[[Y - (1, X, S)×(Y | R, S, T, X) | T = 1, R = 1, X, S]| X, T = 1]] 
    
    =   [[R/r^*(1, X, S)Y - (1, X, S)×(Y | R, S, T, X) | X, T = 1]] 
    
    =   [TR/e^*(X)r^*(1, X, S)Y - (1, X, S)×(Y | R, S, T, X)] 
    
    =   [TR/e^*(X)r^*(1, X, S)Y - (1, X, S)×(Y, R, S, T, X)]

Putting these three terms together, we obtain <Ref>. 

Finally, we can show that ψ_1(Y, R, S, T, X) belongs to the tangent space Λ. 
We can write 

    ψ_1(Y, R, S, T, X) 
           = ψ_1(Y, R, S, T, X) | X
       + ψ_1(Y, R, S, T, X) | T, X - ψ_1(Y, R, S, T, X) | X
       + ψ_1(Y, R, S, T, X) | S, T, X - ψ_1(Y, R, S, T, X) | T, X
       + ψ_1(Y, R, S, T, X) | R, S, T, X - ψ_1(Y, R, S, T, X) | S, T, X
       + ψ_1(Y, R, S, T, X) - ψ_1(Y, R, S, T, X) | R, S, T, X.

It is trivial to show that the terms in the first four lines of the right hand side above belong to 
Λ_X, Λ_T | X, Λ_S | T, X, Λ_R | S, T, X respectively. 
Moreover, the term in the last line 

    ψ_1(Y, R, S, T, X) - ψ_1(Y, R, S, T, X) | R, S, T, X = R ×T/e^*(X)r^*(1, X, S)Y - (1, X, S).

It is easy to show that this term belongs to Λ_Y | R, S, X, T. Therefore, ψ_1(Y, R, S, T, X) belongs to the tangent space Λ, and thus it is the efficient influence function for ξ_1^*.

Similarly, we can show that the efficient influence function for ξ_0^* is 

    ψ_0(Y, R, S, T, X) =    (0, X) - ξ_0^* + 1-T/1 - (X)((0, X, S) - (0 , X)) 
       + (1-T)R/1 - (X)(0, X, S)(Y - (0, X, S))).

It follows that the efficient influence function for δ^* is ψ = ψ_1 - ψ_0, which proves the asserted conclusion in <ref>. 



Under <Ref>, the efficiency lower bound in <Ref> is 

    V^* = [ψ^2(W; δ^*, η^*) ] 
           =  {(TR/(X)(1, X, S) - (1- T)R/(1 - (X))(0, X, S))Y}
       - {(√(1 - e^*(X)/e^*(X))(1, X) + √((X)/1 - (X))(0, X))^2}
       - {1/(X)1 - (1, X, S(1))/(1, X, S(1))(1, X, S(1)) 
                + 1/1 - (X)1 - (0, X, S(0))/(0, X, S(0))(0, X, S(0))}



By straightforward algebra, we can show that  

    V^* = [ψ(W; ξ_1^*, η^*)]  
       = {(TR/(X)(1, X, S) - (1- T)R/(1 - (X))(0, X, S))Y}
       + {(1, X) - (0, X)}_V_1 + {(TR/(X)(1, X, S) - (1- T)R/(1 - (X))(0, X, S))(T, X, S)}_V_2
       + {(T/(X) - 1 - T/1 - (X))(T, X, S)}_V_3 + {(T/(X) - 1 - T/1 - (X))(T, X)}_V_4
       + 2((1, X) - (0, X), (TR/(X)(1, X, S) - (1- T)R/(1 - (X))(0, X, S))(Y - (T, X, S))_V_5
       + 2((1, X) - (0, X), (T/(X) - 1 - T/1 - (X))((T, X, S) - (T, X)))_V_6
       + 2[                  {(T/(X) - 1 - T/1 - (X))((T, X, S) - (T, X)),; (TR/(X)(1, X, S) - (1- T)R/(1 - (X))(0, X, S))(Y - (T, X, S) } ]_V_7
       - 2[        {(TR/(X)(1, X, S) - (1- T)R/(1 - (X))(0, X, S))Y,; (TR/(X)(1, X, S) - (1- T)R/(1 - (X))(0, X, S))(T, X, S)} ]_V_8
       - 2{(T/(X) - 1 - T/1 - (X))(T, X, S), (T/(X) - 1 - T/1 - (X))(T, X)}_V_9

Now we compute these terms one by one. For V_1:

    {(1, X) - (0, X)}   = [((1, X) - (0, X))^2] - ^2[(1, X) - (0, X)] 
       = [((1, X) - (0, X))^2] - (ξ_1^* - ξ_0^*)^2

For V_2 ∼ V_4:

    V_2    = {(TR/(X)(1, X, S) - (1- T)R/(1 - (X))(0, X, S))(T, X, S)}
       = {TR/(X)(1, X, S(1))(1, X, S(1))} + {(1 - T)R/(1 - (X))(0, X, S(0))(0, X, S(0))}
       - 2(TR/(X)(1, X, S(1))(1, X, S(1)), (1 - T)R/(1 - (X))(0, X, S(0))(0, X, S(0))) 
       = {1/(X)(1, X, S(1))(1, X, S(1))} + {1/(1 - (X))(0, X, S(0))(0, X, S(0))}
       - (
        ξ_1^* - ξ_0^*)^2.

since  

    {TR/(X)(1, X, S(1))(1, X, S(1))}   = {[TR | X, S(1)]/((X)(1, X, S(1)))^2(1, X, S(1))}
       - ^2{[TR | X, S(1)]/(X)(1, X, S(1))(1, X, S(1))}
       = {1/(X)(1, X, S(1))(1, X, S(1))} - ξ_1^*2, 
    {(1 - T)R/(1 - (X))(0, X, S(0))(0, X, S(0))}   =  {1/( 1- (X))(0, X, S(0))(0, X, S(0))} - ξ_0^*2,

and 

    (TR/(X)(1, X, S(1))(1, X, S(1)), (1 - T)R/( 1- (X))(0, X, S(0))(0, X, S(0))) 
       = -[TR/(X)(1, X, S(1))(1, X, S(1))][(1 - T)R/( 1- (X))(0, X, S(0))(0, X, S(0)) ] = -ξ_1^*ξ_0^*.

Similarly, 

    V_3    = {1/(X)(1, X, S(1))} + {1/1 - (X)(0, X, S(0))} - (
        ξ_1^* - ξ_0^*)^2 
    
    V_4    = {1/(X)(1, X)} + {1/1 - (X)(0, X)} - (
        ξ_1^* - ξ_0^*)^2.

For V_5 ∼ V_7:

    V_5    = ((1, X) - (0, X), (TR/(X)(1, X, S) - (1- T)R/(1 - (X))(0, X, S))(Y - (T, X, S))) 
       = {((1, X) - (0, X))(TR/(X)(1, X, S) - (1- T)R/(1 - (X))(0, X, S))
       ×([Y | R = 1, T, X, S] - (T, X, S))}
       - {((1, X) - (0, X))}{(TR/(X)(1, X, S) - (1- T)R/(1 - (X))(0, X, S))
       ×([Y | R = 1, T, X, S] - (T, X, S))} = 0,

since [Y | R = 1, T, X, S] = (T, X, S). Similarly V_7 = 0. It is analogous to prove that V_6 = 0 by noting that [(T, X, S) | X, T] = (T, X) according to <ref>.

For V_8 and V_9:

    V_8    = {(TR/(X)(1, X, S) - (1- T)R/(1 - (X))(0, X, S))^2Y(T, X, S)}
       - {(TR/(X)(1, X, S) - (1- T)R/(1 - (X))(0, X, S))(T, X, S)}
       ×{(TR/(X)(1, X, S) - (1- T)R/(1 - (X))(0, X, S))Y}
       = {(TR/(X)(1, X, S) - (1- T)R/(1 - (X))(0, X, S))^2(T, X, S)}
       - ^2{(TR/(X)(1, X, S) - (1- T)R/(1 - (X))(0, X, S))(T, X, S)}
       = {(TR/(X)(1, X, S) - (1- T)R/(1 - (X))(0, X, S))(T, X, S)} = V_2,

where the second equality holds because [Y | R, T, X, S] = (T, X, S). Analogously, we can prove that V_9 = V_4 by again noting that [(T, X, S) | X, T] = (T, X) according to <ref>.

Therefore, 

    V^* 
           = {(TR/(X)(1, X, S) - (1- T)R/(1 - (X))(0, X, S))Y}  + V_1 + V_3 - V_2 - V_4 
       =  {(TR/(X)(1, X, S) - (1- T)R/(1 - (X))(0, X, S))Y}
       - {(√(1 - e^*(X)/e^*(X))(1, X) + √((X)/1 - (X)(0, X)))^2}
       - {1/(X)1 - (1, X, S(1))/(1, X, S(1))(1, X, S(1)) + 1/1 - (X)1 - (0, X, S(0))/(0, X, S(0))(0, X, S(0))},

since 

    V_1 - V_4 
           = -{1 - (X)/(X)(1, X) + (X)/1 - (X)(0, X) + 2(1, X)(0, X)}
       = -{(√(1 - e^*(X)/e^*(X))(1, X) + √((X)/1 - (X)(0, X)))^2},

and 

    V_3 - V_2 = - {1/(X)1 - (1, X, S(1))/(1, X, S(1))(1, X, S(1)) + 1/1 - (X)1 - (0, X, S(0))/(0, X, S(0))(0, X, S(0))}.




We derive the efficient influence functions and efficiency bounds for different settings respectively. In all parts, we focus on efficient influence function for ξ_1^*. The efficient influence function for ξ_0^* and δ^* can be derived analogously. 



  
Efficienct influence function in setting I.
Consider the following model: 

    ℳ_np, I   = {f_X, T, R, Y(X, T, R, Y) = f_X(X)[e(X)^T(1 - e(X))^1 - T][r(T, X)^R(1 - r(T, X))^1- R]
                               × f^R_Y | R = 1, T, X(Y, R, T, X): 
                     f_X  and  f_Y | R = 1, T, Xare arbitrary density functions,
                   and  e(X), r(T, X)  are arbitrary functions obeying  e(X) ∈ [ϵ, 1- ϵ], r(T, X) ∈ [ϵ, 1]  }.

The corresponding tangent space is Λ_I = Λ_X ⊕Λ_T | X⊕Λ_R | T, X⊕Λ_Y | R, T, X, where Λ_X and Λ_T | X are given in the proof of <Ref> and Λ_R | T, X,  Λ_Y | R, T, X are mean square closures of the following sets:

    Λ_R | X, T   = {_R| X, T(R, X, T) ∈ L_2(R, X, T): [_R | X, T(R, X, T) | X, T] = 0}, 
    Λ_Y | R, X, T   = {R ×_Y | R = 1, X, T(Y, R, X, T) ∈ L_2(Y, R, X, T): 
                       [_Y | R = 1, X, T(Y, R, X, T) | R = 1, X, T] = 0}.
 
We again derive the influence function of ξ_1^* = Y(1), which can be written asξ_1^* = Y | X, T = 1, R = 1 according to <Ref>.
Consider regular parametric submodels indexed by γ with score function 

    (Y, R, T, X) = ( X) + (T | X) 
    + (R | T, X) + 
    (Y | R,  T, X),

where the components above satisfy the restrictions imposed in the sets  Λ_X, Λ_T | X, Λ_R | T, X, Λ_Y | R, T, X respectively.
The corresponding target parameter is 

    _γY(1) = _γ_γY | X, T = 1, R = 1.

Then 

    ∂/∂_γY(1)|_γ=0 = ∂/∂_γY | X, T = 1, R = 1|_γ=0 +  ∂/∂_γY | X, T = 1, R = 1|_γ=0.

We have 

    ∂/∂_γY | X, T = 1, R = 1|_γ=0 = (1, X)(X) = (1, X) - ξ_1^*(Y, R, T, X)

and 

    ∂/∂_γY | X, T = 1, R = 1|_γ=0   = Y ×(Y | X, T, R) | X, T = 1, R = 1
       = TR/(X)(1, X)Y - (1, X)(Y, X, T, R).

It follows that 

    ∂/∂_γY(1)|_γ=0 = ψ_I, 1(Y, X, T, R)(Y, X, T, R),
 
where 

    ψ_I, 1(Y, X, T, R) = TR/(X)(1, X)Y - (1, X) + (1, X) - ξ_1^*.

Finally, we can show that ψ_I, 1(Y, R, T, X) belongs to the tangent space Λ_I. 
We can write 

    ψ_I, 1(Y, R, T, X) 
           = ψ_I, 1(Y, R, T, X) | X
       + ψ_I, 1(Y, R, T, X) | T, X - ψ_I, 1(Y, R, T, X) | X
       + ψ_I, 1(Y, R, T, X) | R, T, X - ψ_I, 1(Y, R, T, X) | T, X
       + ψ_I, 1(Y, R, T, X) - ψ_I, 1(Y, R, T, X) | R, T, X.

It is easy to verify that the terms on the right hand side of the equation above belong toΛ_X, Λ_T | X, Λ_R | T, X, Λ_Y | R, T, X respectively. 
Thus ψ_I, 1(Y, R, T, X) belongs to the tangent space Λ_I. 
This shows that ψ_I, 1 is the efficient influence function of ξ_1^*.
Similarly, we can derive the efficient influence function ψ_I, 0 for ξ_0^* as follows: 

    ψ_I, 0(Y, R, T, X)  = (1-T)R/(X)(0, X)Y - (0, X) + (0, X) - ξ_0^*.

It follows that ψ_I, 1 - ψ_I, 0 is the efficient influence function for δ^*. 




  
Efficienct influence function in setting II. 
Now we consider the model 

    ℳ_np, II   = {f_X, T, R, Y, S(X, T, R, Y) = f_X(X)[e(X)^T(1 - e(X))^1 - T][r(T, X)^R(1 - r(T, X))^1- R]: 
                               × f^R_Y | R = 1, T, X(Y, R, T, X) f^R_S | R = 1, T, X, Y(S, R, T, X, Y), 
                    f_X  and  f_Y | R = 1, T, Xare arbitrary density functions,
                   and  e(X), r(T, X)  are arbitrary functions obeying  e(X) ∈ [ϵ, 1- ϵ], r(T, X) ∈ [ϵ, 1]  }.

The corresponding tangent space is Λ_II = Λ_X ⊕Λ_T | X⊕Λ_R | T, X⊕Λ_Y | R, T, X⊕Λ_S | Y, R, T, X, where Λ_X and Λ_T | X are given in the proof of <Ref> and Λ_R | T, X,  Λ_Y | R, T, X, Λ_S | Y, R, T, X are mean square closures of the following sets:

    Λ_R | X, T   = {_R| X, T(R, X, T) ∈ L_2(R, X, T): [_R | X, T(R, X, T) | X, T] = 0}, 
    Λ_Y | R, X, T   = {R ×_Y | R = 1, X, T(Y, R, X, T) ∈ L_2(Y, R, X, T): 
                       [_Y | R = 1, X, T(Y, R, X, T) | R = 1, X, T] = 0}
    Λ_S | Y, R, X, T   = {R ×_S | Y, R = 1, X, T(S, Y, R, X, T) ∈ L_2(S, Y, R, X, T): 
                       [_S | Y, R = 1, X, T(S, Y, R, X, T) | Y, R = 1, X, T] = 0}.
 
 Consider regular parametric submodels indexed by γ with score function 

    (S, Y, R, T, X) = ( X) + (T | X) 
    + (R | T, X) + 
    (Y | R,  T, X) + (S | Y, R, T, X),

where the components above satisfy the restrictions imposed in the sets  Λ_X, Λ_T | X, Λ_R | T, X, Λ_Y | R, T, X, Λ_S | Y, R, T, X respectively.
The corresponding target parameter is 

    _γY(1) = _γ_γY | X, T = 1, R = 1.

The analyses for setting I already shows that 

    ∂/∂_γY(1)|_γ=0 = ψ_I, 1(Y, X, T, R)(Y, X, T, R),
 
 It follows that 
 
    ∂/∂_γY(1)|_γ=0 = ψ_I, 1(Y, X, T, R)(S, Y, R, T, X),
 
 since
 
    ψ_I, 1(Y, X, T, R)(S | Y, R, T, X) = ψ_I, 1(Y, X, T, R)(S | Y, R, T, X)| Y, R, T, X = 0.
 
  This means that ψ_I, 1 is also an influence function for ξ_1^* under the model ℳ_np, II.

  Moreover, we showed that ψ_I, 1∈Λ_I. 
  Since Λ_I ⊂Λ_II, we also have ψ_I, 1∈Λ_II.
  It follows that ψ_I, 1 is also the efficient influence function for ξ_1^* under the model ℳ_np, II.
  Similarly, we can validate that 
ψ_I, 1 - ψ_I, 0 is the efficient influence function of δ^* udner the model ℳ_np, II.







  
Efficienct influence function in setting III. 
Under the additional <Ref>, we have R ⊥ S | T, X, thus the tangent space under <Ref> now becomes 

    Λ_III = Λ_X ⊕Λ_T | X⊕Λ_S | T, X⊕Λ_R | T, X⊕Λ_Y | R, S, T, X,

where Λ_X, Λ_T | X, Λ_S | T, X, Λ_Y | R, S, T, X are given in the proof for <Ref>, and Λ_R | T, X is the mean-square closure of the set 

    Λ_R | X, T   = {_R| X, T(R, X, T) ∈ L_2(R, X, T): [_R | X, T(R, X, T) | X, T] = 0}.


The function ψ in <Ref> is again an influence function in setting III with the additional <Ref>. 
Moreover, it is easy to show that 

    (1, X) - (0, X) - δ^* ∈Λ_X ⊕Λ_T | X
       T/(X)((1, X, S) - (1 , X)) - 1 - T/1 - (X)((0, X, S) - (0 , X)) ∈Λ_S | T, X
       TR/(X)(1, X, S)(Y - (1, X, S)) - (1 - T)R/(1 - (X))(0, X, S)(Y - (0, X, S)) ∈Λ_Y | R, S, T, X.


It follows that ψ∈Λ_III. 
Thus ψ is again the efficient influence function. 




  
Efficienct influence function in setting IV.
The efficient influence function in setting IV directly follows from <cit.> so we omit the details. 
  


Under <Ref>, the efficiency lower bounds for setting I-IV in <Ref> are given as follows: 

    V_I^* = V_II^*  
           = {(TR/(X)(1, X) - (1 - T)R/(1 - (X))(0, X))Y}
       - {(R = 0 | X)/(X)(1, X)(1, X) + (R = 0 | X)/(X)(0, X)(0, X)}
       -{(√(1 - e^*(X)/e^*(X)(0, X)/(1, X))(1, X) + √((X)/1 - (X)(1, X)/(0, X))(0, X))^2}
    
    V^*_III   =  {(TR/(X)(1, X) - (1- T)R/(1 - (X))(0, X))Y}
       - {(√(1 - e^*(X)/e^*(X))(1, X) + √((X)/1 - (X))(0, X))^2}
       - {1/(X)1 - (1, X)/(1, X)(1, X, S(1)) + 1/1 - (X)1 - (0, X)/(0, X)(0, X, S(0))}
    
    V_IV^* 
           = {(T/(X) - 1 - T/1 - (X))Y} - {(√(1 - e^*(X)/e^*(X))(1, X) + √((X)/1 - (X))(0, X))^2}.




Efficiency bound in setting I. The semiparametric efficiency bound is given by {ψ_I(W; δ^*, η^*)}:

    V_I^* 
           = {ψ_I(W; δ^*, η^*)}
       ={(TR/(X)(1, X) - (1 - T)R/(1 - (X))(0, X))Y} + {(1, X) - (0, X)}_V_10
       + {(TR/(X)(1, X) - (1 - T)R/(1 - (X))(0, X))(T, X)}_V_11
       - 2[     {(TR/(X)(1, X) - (1 - T)R/(1 - (X))(0, X))Y,; (TR/(X)(1, X) - (1 - T)R/(1 - (X))(0, X))(T, X)} ]_V_12
       - 2{(1, X) - (0, X), (TR/(X)(1, X) - (1 - T)R/(1 - (X))(0, X))(T, X)}_V_13
       + 2{(1, X) - (0, X), (TR/(X)(1, X) - (1 - T)R/(1 - (X))(0, X))Y}_V_14

Similarly to Step IV in the proof of <Ref>, we can show that 

    V_10   = [(1, X) - (0, X)]^2 - (ξ_1^* - ξ_0^*)^2
     
    V_11 = V_12   = {^2(1, X)/(X)(1, X) + ^2(0, X)/(X)(0, X)} - (ξ_1^* - ξ_0^*)^2 
    
    V_13   = V_14.

Therefore,

    V_I^*  = {ψ_I(W; δ^*, η^*)}   = {(TR/(X)(1, X) - (1 - T)R/(1 - (X))(0, X))Y} + V_10 - V_11
       = {(TR/(X)(1, X) - (1 - T)R/(1 - (X))(0, X))Y}
       - {(√(1 - e^*(X)/e^*(X)(0, X)/(1, X))(1, X) + √((X)/1 - (X)(1, X)/(0, X))(0, X))^2}
       - {(R = 0 | X)/(X)(1, X)(1, X) + (R = 0 | X)/(X)(0, X)(0, X)}.
 
 The second equality above holds because
 
    V_10 - V_11   =  [(1, X) - (0, X)]^2 - {^2(1, X)/(X)(1, X) + ^2(0, X)/(X)(0, X)}
       = -{1 - (T = 1, R = 1 | X)/(T = 1, R = 1 | X)^2(1, X) + 1 - (T = 0, R = 1 | X)/(T = 0, R = 1 | X)^2(0, X)}
               - 2{(1, X)(0, X)}
       = -{(R = 1 | X) - (T = 1, R = 1 | X)/(T = 1, R = 1 | X)^2(1, X) 
            + (R = 1 | X) - (T = 0, R = 1 | X)/(T = 0, R = 1 | X)^2(0, X)}
       - {(R = 0 | X)/(T = 1, R = 1 | X)^2(1, X) + (R = 0 | X)/(T = 0, R = 1 | X)^2(0, X)} - 2{(1, X)(0, X)}
       = -{(1 - (X))(0, X)/(X)(1, X)^2(1, X) + (X)(1, X)/(1 - (X))(0, X)^2(0, X)} - 2{(1, X)(0, X)}
       - {(R = 0 | X)/(X)(1, X)^2(1, X) + (R = 0 | X)/(1 - (X))(0, X)^2(0, X)}
       = - {(√(1 - e^*(X)/e^*(X)(0, X)/(1, X))(1, X) + √((X)/1 - (X)(1, X)/(0, X))(0, X))^2}
       - {(R = 0 | X)/(X)(1, X)(1, X) + (R = 0 | X)/(X)(0, X)(0, X)}.


 Efficiency lower bound in setting II. From the proof of <Ref>, we know that V_2^* = V_1^*. 

 Efficiency lower bound in setting III.  The conclusion follows directly from <Ref> by noting that (t, X, S) = (t, X).

 Efficiency lower bound in setting IV. The efficiency lower bound is given by {ψ^2_IV(W; δ^*, η^*)}:

    V_IV^* = {ψ^2_IV(W; δ^*, η^*)}
       = {(T/(X) - 1 - T/1 - (X))Y} + {(1, X) - (0, X) }
       +  {(T/(X) - 1 - T/1 - (X))(T, X)}
       + 2{(T/(X) - 1 - T/1 - (X))Y, (1, X) - (0, X)}
       - 2{(T/(X) - 1 - T/1 - (X))(T, X), (1, X) - (0, X)}
       - 2{(T/(X) - 1 - T/1 - (X))Y, (T/(X) - 1 - T/1 - (X))(T, X)}.

Analogously to step IV in the proof of <Ref>, we can show that 

    {(T/(X) - 1 - T/1 - (X))(T, X)}
               = {(T/(X) - 1 - T/1 - (X))Y, (T/(X) - 1 - T/1 - (X))(T, X)}
                = {1/(X)^2(1, X) + 1/1 - (X)^2(0, X)} - (λ^*_1 - λ^*_0)^2 
       {(T/(X) - 1 - T/1 - (X))Y, (1, X) - (0, X)}
                = {(T/(X) - 1 - T/1 - (X))(T, X), (1, X) - (0, X)}.

Thus 

    {ψ_IV(W; δ^*, η^*)}
       = {(T/(X) - 1 - T/1 - (X))Y} - {1/(X)^2(1, X) + 1/1 - (X)^2(0, X)}
       + [(1, X) - (0, X)]^2 
       = {(T/(X) - 1 - T/1 - (X))Y} - {(√(1 - e^*(X)/e^*(X))(1, X) + √((X)/1 - (X))(0, X))^2}.




According to <ref> and <ref>, we can verify that 

    V_I^* - V^*_III  = V_II^* - V^*_III
       ={(√(1 - e^*(X)/e^*(X))(1, X) + √((X)/1 - (X))(0, X))^2}
       - {(R = 0 | X)/(X)(1, X)(1, X) + (R = 0 | X)/(X)(0, X)(0, X)}
            -{(√(1 - e^*(X)/e^*(X)(0, X)/(1, X))(1, X) + √((X)/1 - (X)(1, X)/(0, X))(0, X))^2}
       + {1/(X)1 - (1, X)/(1, X)(1, X, S(1)) + 1/1 - (X)1 - (0, X)/(0, X)(0, X, S(0))}
       = {1 - e^*(X)/e^*(X)^2(1, X) + e^*(X)/1 - e^*(X)^2(0, X) } +  2{(1, X)(0, X)}
            -{1 - (T = 1, R = 1 | X)/(T = 1, R = 1 | X)^2(1, X) + 1 - (T = 0, R = 1 | X)/(T = 0, R = 1 | X)^2(0, X)}
            - 2{(1, X)(0, X)}
            + {1/(X)1 - (1, X)/(1, X)(1, X, S(1)) + 1/1 - (X)1 - (0, X)/(0, X)(0, X, S(0))}
       =  {1 - e^*(X)/e^*(X)^2(1, X) + e^*(X)/1 - e^*(X)^2(0, X) }
           - {1 - (X)(1, X)/(X)(1, X)^2(1, X) + 1 - (1 - (X))(0, X)/(1 - (X))(0, X)^2(0, X)}
           + {1/(X)1 - (1, X)/(1, X)(1, X, S(1)) + 1/1 - (X)1 - (0, X)/(0, X)(0, X, S(0))}
       = {1/(X)1 - (1, X)/(1, X)((1, X, S(1)) - (1, X)) 
           + 1/1 - (X)1 - (0, X)/(0, X)((0, X, S(0)) - (0, X))}

Then the conclusion in statement 1 follows from the fact that (t, x) = [Y | T = t, X = x,R = 1] under <Ref> according to <Ref>.

Furthermore, 

    V_III^* - V^*_IV
       = {(TR/(X)(1, X) - (1- T)R/(1 - (X))(0, X))Y} - {(T/(X) - (1- T)/(1 - (X)))Y}
       - {1/(X)1 - (1, X)/(1, X)(1, X, S(1)) + 1/1 - (X)1 - (0, X)/(0, X)(0, X, S(0))}
       = {1/(X)(1, X)Y^2(1) + 1/(X)(0, X)Y^2(0)} - (ξ_1^* - ξ_0^*)^2 - {1/(X)Y^2(1) + 1/(X)Y^2(0)}
                    + (ξ_1^* - ξ_0^*)^2  
       - {1/(X)1 - (1, X)/(1, X)(1, X, S(1)) + 1/1 - (X)1 - (0, X)/(0, X)(0, X, S(0))}
       = {1/(X)1 - (1, X)/(1, X)(Y^2(1) - (1, X, S(1))) + 1/1 - (X)1 - (0, X)/(0, X)(Y^2(0) - (0, X, S(0)))}.

This obviously implies the conclusion in statement 2. 



Note that 

    Y | S, X, T = t, R = 1| X, T =t, R = 0| R = 0
    
    =   Y(t) | S(t), X, T = t, R = 1| X, T = t, R = 0| R = 0
    
    =   Y(t) | S(t), X, T = t, R = 0| X, T = t, R = 0| R = 0<Ref>
    
    =   Y(t) | S(t), X, R = 0| X, R = 0| R = 0Y(t), S(t)⊥ T | X, R = 0
    
    =   Y(t) | R = 0.

This proves the identification of δ_0^*. 

Now we derive the efficiency bound for Y(1) | R = 0, based on the ℳ_np model and the corresponding parametric submodels in the proof for <Ref>.
Consider the target parameter under parametric submodels indexed by parameters γ. Then 

    ∂/∂γ_γY(1) | R = 0|_γ = 0
    
       =   ∂/∂γ_γ_γ_γY | S, X, T = 1, R = 1| X, T =1, R = 0| R = 0|_γ = 0
    
       =   ∂/∂γ_γ_0(1, X) | R = 0|_γ = 0 + ∂/∂γ_γ(1, X, S)| X, T=1,R=0| R = 0|_γ = 0
    
       +   ∂/∂γ_γY | S, X, T = 1, R = 1| X, T =1, R = 0| R = 0|_γ = 0.

Again, we deal with each term respectively. 

First of all, 

    ∂/∂γ_γ_0(1, X) | R = 0|_γ = 0 = 1-R/R=0_0(1, X) - Y(1) | R = 0(Y, R, S, T, X).


Secondly, 

    ∂/∂γ_γ(1, X, S)| X, T=1,R=0| R = 0|_γ = 0
    
    =   (1, X, S)(S | X, T, R) | X, T = 1, R = 0| R = 0
    
    =   (1, X, S) - (1, X)(S | X, T, R) | X, T = 1, R = 0| R = 0
    
    =   (1, X, S) - (1, X)(Y, S, X, T, R) | X, T = 1, R = 0| R = 0
    
    =   T(1-R)/T=1| R=0, XR= 0(1, X, S) - _0(1, X)(Y, S, X, T, R).


Thirdly, 

    ∂/∂γ_γY | S, X, T = 1, R = 1| X, T =1, R = 0| R = 0|_γ = 0
    
    =   Y ×(Y | S, X, T, R) | S, X, T = 1, R = 1| X, T =1, R = 0| R = 0
    
    =   Y - (1, X, S)×(Y | S, X, T, R) | S, X, T = 1, R = 1| X, T =1, R = 0| R = 0
    
    =   Y - (1, X, S)×(Y, S, X, T, R) | S, X, T = 1, R = 1| X, T =1, R = 0| R = 0
    
    =   [[R/R=1| S, X, T=1R=0| S, X, T=1/R= 0| X, T=1Y - (1, X, S)
                   ×(Y, S, X, T, R) | X, T = 1] | R = 0] 
    
    =   𝔼[R/R=1| S, X, T=1R=0| S, X, T=1/R= 0| X, T=1T/T=1| XR= 0 | X/R=0
                   ×Y - (1, X, S)(Y, S, X, T, R)] 
    
    =   R/R=1| S, X, T=1T/T=1| R=0,XR=0| S, X, T=1/R= 0Y - (1, X, S)(Y, S, X, T, R).


Putting the three equations above together, we have 

    ∂/∂γ_γY(1) | R = 0|_γ = 0= ψ̃_1(W)(Y, S, X, T, R),

where 

    ψ̃_1(W) 
          = 1-R/R=0_0(1, X) - Y(1) | R = 0 + T(1-R)/R= 0T=1| R=0, X(1, X, S) - _0(1, X)
       + TR/R= 0T=1| R=0,XR=0| S, X, T/R=1| S, X, TY - (1, X, S).

We can also use the decomposition in the proof for <Ref> to show that ψ̃_1 belongs to the tangent space, so it is also the efficient influence function for Y(1) | R=0.
We can similarly derive the efficient influence function for Y(0) | R=0.
The final efficient influence function for δ_0^*  is 

    1-R/R=0_0(1, X) - _0(0, X) - δ_0^*
    
    +   1-R/R=0{T/T=1| R=0, X(1, X, S) - _0(1, X)
       - 1- T/1 - T=1| R=0, X(0, X, S) - _0(0, X)}
    
    +   R/R=0R=0| S, X, T/R=1| S, X, T×
           T/T=1| R=0,XY - (1, X, S) - 1-T/1-T=1| R=0,XY - (0, X, S)





 §.§ Proofs for <ref>


Let 

    ψ_1(W; ξ_1, η) = μ(1, X) - ξ_1 + T/e(X)(μ(1, X, S) - μ(1 , X)) + TR/e(X)r(1, X, S)(Y -(1, X, S))), 
       ψ_0(W; ξ_0, η) = μ(0, X) - ξ_0 + 1-T/1 - e(X)((0, X, S) - μ(0 , X)) + (1-T)R/1 - e(X)r(0, X, S)(Y - (0, X, S))).

Then ψ(W; δ, η) = ψ_1(W; ξ_1, η) - ψ_0(W; ξ_0, η) for any δ = ξ_1 + ξ_0. 

By straightforward algebra, we can show that 

    ψ_1(W; ξ_1, η_0)-ψ_1(W; ξ_1, η^*)   = (1-T/(X))(1, X) - μ_0(1, X)
       + T/(X)1-R/(1, X, S)_0(1, X, S) - (1, X, S)
       + T/e_0(X) - T/(X)(1, X) - μ_0(1, X)
       + T/e_0(X)1-R/r_0(1, X, S)_0(1, X, S) - (1, X, S).

Here the first two terms on the right hand side are equal to 0 because T | X = (X) and R | T = 1, X, S = (1, X, S). Moreover, 

    T/e_0(X) - T/(X)(1, X) - μ_0(1, X) = (X) - e_0(X)/e_0(X)(1, X) - μ_0(1, X),

and 

    T/e_0(X)1-R/r_0(1, X, S)_0(1, X, S) - (1, X, S)   = 𝔼[(X)r_0(1, X, S) - (1, X, S)/e_0(X)r_0(1, X, S)
           ×_0(1, X, S) - (1, X, S)].

Similarly, we can derive ψ_1(W; ξ_1, η_0)-ψ_1(W; ξ_1, η^*) and obtain 

    ψ(W; δ, η_0)-ψ(W; δ, η^*)   ≤1/ϵ - e_0_2(1, X) - μ_0(1, X)_2 + (0, X) - μ_0(0, X)_2
       + 1/ϵ^2r_0(1, X, S) - (1, X, S)_2_0(1, X, S) - (1, X, S)_2 
       + 1/ϵ^2r_0(0, X, S) - (0, X, S)_2_0(0, X, S) - (0, X, S)_2.

By following the proof of <Ref>, we have

    max_0(0, X, S) - (0, X, S)_2, _0(1, X, S) - (1, X, S)_2≤1/√(2ϵ)_0(T, X, S) - (T, X, S)_2, 
       maxr_0(0, X, S) - (0, X, S)_2, r_0(1, X, S) - (1, X, S)_2≤1/√(2ϵ)r_0(T, X, S) - (T, X, S)_2.
 
 Therefore, 
 
    ψ(W; δ, η_0)-ψ(W; δ, η^*)≤√(2)ϵ^-3/2e_0 - _2μ_0 - _2 + ϵ^-3r_0 - _2_0 - _2.




With slight abuse of notation, we define the following function for  η = e, r, μ,: 

    ψ(W; η)
          =  μ(1, X) - μ(0, X) - δ^* 
       =+ TR/e(X)r(1, X, S)(Y - (1, X, S)) - (1 - T)R/(1 - e(X))r(0, X, S)(Y - (0, X, S)) 
       =+ T/e(X)((1, X, S) - μ(1 , X)) - 1 - T/1 - e(X)((0, X, S) - μ(0 , X)).

Then our estimator is 

    δ̂= 1/K∑_k=1^K [ψ(W; η̂_k)].


We can decompose the estimation error of δ̂ as follows

    δ̂- δ^*   = 1/K∑_k=1^K[ψ(W; η̂_k)] - ψ(W; η̂_k)|η̂_k - [ψ(W; η_0)] - [ψ(W; η_0)]_ℛ_1, k
       + 1/K∑_k=1^Kψ(W; η̂_k)|η̂_k - [ψ(W; η_0)]_ℛ_2, k + 1/N∑_i=1^n ψ(W_i; η_0) - δ^*_ℛ_3,

where η_0 is the limit of η̂_k for k = 1, …, K as N →∞.

Here we can easily show that as N →∞, η̂_k - η_0→ 0, we have 

    Varℛ_1, k|η̂_k→ 0.

Moreover, we have ℛ_2, k→ 0 because η̂_k - η_0→ 0. 
Finally, by law of large number, we have 

    ℛ_3→ψ(W_i; η_0) - δ^* = ψ(W_i; η_0) - ψ(W_i; η^*).
 
 According to <Ref>, the last display is equal to 0 as long as 
 
    ( - )( - ) = 0,    ( - )( - ) = 0.
 
 Therefore, δ̂→δ^* as N →∞ as long as ( - )( - ) = 0 and ( - )( - ) = 0.



We start with the error decomposition in the proof for  <ref> with η_0 = η^*. 
As we mentioned there, 

    Varℛ_1, k|η̂_k→ 0.

So by Chebyshev's inequality, we have ℛ_1, k = o_p(N^-1/2). 

Moreover, by <Ref>, we have 

    ℛ_2, k   ≤√(2)ϵ^-3/2ê_k - _2μ̂_k - _2 + ϵ^-3r̂_k - _2_k - _2 
       ≤√(2)ϵ^-3/2 + ϵ^-3 = O_pρ_N, eρ_N, μ + ρ_N, rρ_N, .

So  under the conditions that = o(N^-1/2), 

= o(N^-1/2), we have ℛ_2, k = o_p(N^-1/2). 

Therefore,

    √(N)δ̂- δ^*  = 1/√(N)∑_i=1^n ψ(W_i; η^*) - δ^* + o_p(1).

Then the asserted conclusion follows from  central limit theorem. 







Note that 

    |V̂ -  V^*| = |V̂ -  [ψ^2(W; δ^*, η^*)]| = 1/K∑_k = 1^K|[ψ^2(W; δ̂, η̂_k)] - [ψ^2(W; δ^*, η^*)]|,

We only need to prove that |[ψ^2(W; δ̂, η̂_k)] - [ψ^2(W; δ^*, η^*)]| = o_p(1). Consider the following decomposition:

    |[ψ^2(W; δ̂, η̂_k)] - [ψ^2(W; δ^*, η^*)]|  
           ≤ |[ψ^2(W; δ̂, η̂_k)] - [ψ^2(W; δ^*, η^*)]| 
       + |[ψ^2(W; δ^*, η^*)] - [ψ^2(W; δ^*, η^*)]| 
       = ℛ_4 + ℛ_5.

Thus we only need to prove that both ℛ_4 and ℛ_5 are o_p(1).



  
*Bounding ℛ_5.  According to <Ref>, 

    ψ(W; δ^*, η^*)_q 
           ≤1/ϵ^2(Y(1)_q + Y(0)_q) + 1 + ϵ/ϵ^2((1, X, S(1))_q + (0, X, S(0))_q) 
       + 1 + ϵ/ϵ ((1, X)_q + (0, X)_q) ≤[2/ϵ^2 + 2(1 + ϵ)/ϵ^2 + 2(1 + ϵ)/ϵ]C.



If q in <ref> satisfies that q ≥ 4, then 

    [ℛ_5^2] = 1/n{ψ^2(W; δ^*, η^*)}≤1/n{ψ^4(W; δ^*, η^*)}.

By Markov inequality, ℛ_5 = O_p(N^-1/2).

If 2 < q < 4, then we apply the von Bahr-Esseen inequality with p = q/2 ∈ (1, 2): 

    [ℛ_5^q/2] ≤ 2n^-q/2 + 1[ψ^q(W; δ^*, η^*)] = 2n^-q/2 + 1ψ(W; δ^*, η^*)_q^q.




Thus [ℛ_5^q/2] = O(2N^-q/2 + 1), which implies that ℛ_5 = O_p(N^-1 + 2/q) according to Markov inequality.

Therefore 
    ℛ_5 = O_p(N^-[(1 - 2/q) ∨ 1/2]) = o_p(1).




  
*Bounding ℛ_4. Simple algebra shows that for any a, δ a, 

    (a + δ a)^2 - a^2 = δ a(2a + δ a).

Now take a + δ a = ψ(W; δ̂, η̂_k) and a = ψ(W; δ^*, η^*), then

    |[ψ^2(W; δ̂, η̂_k)] - [ψ^2(W; δ^*, η^*)]| 
    
    =   |(ψ(W; δ̂, η̂_k) - ψ(W; δ^*, η^*))(2ψ(W; δ^*, η^*) + ψ(W; δ̂, η̂_k) - ψ(W; δ^*, η^*))| 
    ≤   ([ψ(W; δ̂, η̂_k) - ψ(W; δ^*, η^*)]^2)^1/2([2ψ(W; δ^*, η^*) + ψ(W; δ̂, η̂_k) - ψ(W; δ^*, η^*)]^2)^1/2
    ≤   ℛ_6^1/2× (ℛ_6^1/2 + 2( [ψ^2(W; δ^*, η^*)])^1/2).

where ℛ_6 =[ψ(W; δ̂, η̂_k) - ψ(W; δ^*, η^*)]^2.


Since [ψ^2(W; δ^*, η^*)] = O(1), Markov inequality implies that [ψ^2(W; δ^*, η^*)] = O_p(1).

Moreover,

    ℛ_6≤ 2 (δ̂ - δ^*)^2 + 2[ψ(W; δ^*, η̂_k) - ψ(W; δ^*, η^*)]^2.

Since δ̂ - δ^* = o_p(1) according to <ref>, thus we only need to prove [ψ(W; δ^*, η̂_k) - ψ(W; δ^*, η^*)]^2 = o_p(1) as well. 
We can further decompose this term: 

    [ψ(W; δ^*, η̂_k) - ψ(W; δ^*, η^*)]^2 
          ≤|ψ(W; δ^*, η̂_k) - ψ(W; δ^*, η^*)^2
                -ψ(W; δ^*, η̂_k) - ψ(W; δ^*, η^*)^2 |η̂_k| 
       + ψ(W; δ^*, η̂_k) - ψ(W; δ^*, η^*)^2 |η̂_k.

Note that 

    ψ(W; δ^*, η̂_k) - ψ(W; δ^*, η^*)^2 - ψ(W; δ^*, η̂_k) - ψ(W; δ^*, η^*)^2 |η̂_k|η̂_k = 0,

so  by Markov inequality, we have 

    ψ(W; δ^*, η̂_k) - ψ(W; δ^*, η^*)^2 - ψ(W; δ^*, η̂_k) - ψ(W; δ^*, η^*)^2 |η̂_k = o_p(1).

Moreover, it is easy to verify that ψ(W; δ^*, η̂_k) - ψ(W; δ^*, η^*)^2 |η̂_k = o_p(1) as η̂_k - η^* = o_p(1).

Putting all above together, we have 

    ℛ_4 = o_p(1).




  
*Conclusion. 
Therefore, V̂ = V^* + o_p(1), and by Slutsky's theorem, 

    √(N)(δ̂  - δ^*)/√(V̂)d→𝒩(0, 1),

so that (δ^* ∈CI) → 1 - α.





 §.§ Proofs for <ref>


We prove a more general conclusion without assuming R ⊥ T | X. The asserted conclusion in this theorem then follows straightforwardly. Note that here we only observe the labeled data, so all distributions of the observed data are  conditional distributions given R = 1. These conditional distributions, as we assumed, are not dependent on the sample size N. Also the ATE is defined in terms of  that does not depend on sample size N as well. 
Thus throughout the proof, all observed data are i.i.d. samples from a fixed distribution that does not vary with sample size N. 
This means that we can directly apply  standard semiparametric efficiency theory, and do not need to customize the efficiency theory to N-dependent data generating process  <cit.>.

Note that we assume the distribution of (X, T, S) is known. According to <Ref> that R ⊥ S(t) | T, X, the distribution of S given T, X and R = 1 is also known: for any s ∈𝒮 and x ∈𝒳,

    (S ≤ s | X = x, T = t, R = 1) 
           = (S(t) ≤ s | X = x, T = t, R = 1)  
       = (S(t) ≤ s | X = x, T = t) 
       = (S ≤ s | X = x, T = t).

We introduce the following notations: we use f^*_S | X, T, R = 1(s, x, t) to denote the density function of the distribution of S given T, X and R = 1. Moreover, we denote (1, x) (T = 1 | R = 1, X = x), (1, x, s) (T = 1 | R = 1, X = x, S(1) = s) for any x ∈𝒳 and s ∈𝒮.

Since we only observe the labeled data, we consider the following model for the distribution of the observed data 

    ℳ̃_np 
            = {   f_X, T, S, Y | R = 1(X, T, S, Y | R = 1) 
                = f_X | R= 1(X | R = 1)[e(1, X)^T(1 - e(1, X))^1 - T]
            f^*_S | X, T, R = 1(S | X, T, R = 1) f_Y | S, T, X, R = 1(Y, S, T, X | R = 1):  ∀ e(1, X) ∈ [ϵ, 1- ϵ], 
             f_X | R = 1,  and  f_Y | S, T, X, R = 1are arbitrary density functions}

First, note that under <Ref> and <Ref> <Ref>, 

    ξ_1^* 
           = [Y(1)] = ∫ f^*_X(x) [Y(1) | X = x]dx  
       = ∬ f^*_X(x)f^*_S | X, T = 1, R = 1(s | x, T = 1, R = 1) [Y | X = x, S = s, T = 1, R = 1]dsdx 
       = ∭ yf^*_X(x)f^*_S | X, T = 1, R = 1(s | x, T = 1, R = 1)f_Y | X, S, T = 1, R = 1(y | s, x, T = 1, R = 1) dydsdx.

Here both f^*_X and f^*_S | X, T = 1, R = 1 are known according to our assumption that we know the distribution of (X, T, S). 
So the only unknown part that requires modeling and contributes to the definition of ξ_1^* is the distribution of Y given S, T, X, R. 

Therefore, the efficiency lower bound of estimating ξ_1^* under model ℳ_np is the same as that under the following model ℳ_fix that fixes all components but the distribution of Y given S, T, X, R = 1 at truth:

    ℳ_fix   = {f_X, T, S, Y | R = 1(X, T, S, Y | R = 1) = f^*_X | R= 1(X | R = 1)[(1, X)^T(1 - (1, X))^1 - T] 
               × f^*_S | X, T, R = 1(S | X, T, R = 1)f_Y | S, T, X, R = 1(Y, S, T, X | R = 1)
               : f_Y | S, T, X, R = 1is an arbitrary density function}.

It is easy to verify that the tangent space of ℳ_fix and its orthogonal complement are

    Λ_fix   = {Y | X, T, S∈ℒ_2(Y, X, T, S): [Y | X, T, S| X, T, S, R = 1] = 0}, 
    Λ^⊥_fix   = {X, T, S∈ℒ_2(X, T, S): [X, T, S| R = 1] = 0}.




Again we consider smooth and regular parametric submodels indexed by γ when analyzing path-differentiability of ξ_1^*. In the following part, we suppress the subscripts in the density functions f^*, and the meaning of the density functions should be self-evident from the arguments. For example, f^*(x) is the density for X. 
We have 

    ∂/∂γ_γ[Y(1)] |_γ = γ^*   =  ∫ f^*(x) f^*(s | X = x, T = 1, R = 1)∂/∂γ_γ[Y | X = x, S = s, T = 1, R = 1] |_γ = 0dxds 
       = ∫ f^*(x) f^*(s | X = x, T = 1, R = 1)[Y ×Y | X, S, T| X = x, S = s, T = 1, R = 1]dxds 
       = ∫ f^*(x) f^*(s | X = x, T = 1, R = 1)
           ×T/(1, X, S)(Y - (T, X, S))Y | X, S, T| X = x, S = s, R = 1dxds 
       = ∫ f^*(x) f^*(s | X = x, T= 1, R = 1)
           ×T/(1, X)(Y - (T, X, S))Y | X, S, T| X = x, S = s, R = 1dxds
    
           = f^*(X)/f^*(X | R = 1)T/(1, X)(Y - (T, X, S))Y | X, S, T| R = 1.

Here the fourth equality holds because under <Ref>, 

    (1, X, S(1)) = (T = 1| R= 1, X, S(1)) 
           = (R = 1 | T = 1, X, S(1))(T = 1 | X, S(1))/∑_t= 0, 1(R = 1 | T = t, X, S(1))(T = t | X, S(1))
       = (R = 1 | T = 1, X)(T = 1 | X)/∑_t= 0, 1(R = 1 | T = t, X)(T = t | X) = (1, X),

and thus 

    T/(1, X, S) = T/(1, X, S(1)) = T/(1, X).




It is easy to verify that f^*(X)/f^*(X | R = 1)T/(1, X)(Y - (T, X, S)) is orthogonal to Λ^⊥_fix, thus it is exactly the efficient influence function for ξ_1^*. Similarly we can prove that the efficient influence function for ξ_0^* is f^*(X)/f^*(X | R = 1)1 - T/1 - (1, X)(Y - (T, X, S)), and the efficient influence function for δ^* is the difference between the efficient influence functions for ξ_1^* and ξ_0^*.

When R ⊥ T | X indeed holds, we know that (1, X) = (0, X) =  (X), which implies the asserted conclusion. 





We prove the result for estimator ξ̂_1^rev  that estimates ξ_1^* = [Y(1)]. We can analogously prove the result for ξ̂_0^rev and δ̂^rev. Note that we allow the data generating process to depend on N and the observations form a triangular array {W_i, N = (X_i, N, T_i, N, S_i, N, R_i, N, Y_i, N): i = 1, …, N}.
We denote the expectation and variance under distribution ^(N) as _N and _N respectively. Note that once conditional on R_i, N = 1, they are the same as expectation and variance under the distribution of  given R = 1.

Define the following function for any r > 0 and η̃= e, λ, μ̃, μ with e(X) ≥ϵ almost surely: 

    ψ_1(W; r, η̃) = μ(1, X) + T/e(X)(1, X, S) - μ(1, X) + λ(X)R/rT/e(X)Y - (1, X, S).

 According to <ref>, the estimator ξ̂_1^rev corresponding to δ̂^rev is 

    ξ̂_1^rev  = 1/K∑_k = 1^K^N_kψ_1(W; r̂_N, η̂̃̂_k).

Similar to the proof for <Ref>, we can show that for any r > 0 and η̃= e, λ, μ̃, μ with e(X) ≥ϵ almost surely and λ(X)_∞≤ C: 

    _Nψ_1(W; r_0, η̃) - ψ_1(W; r_N, η̃^*)   ≤1/ϵe - _2μ(1, X) - (1, X)_2 
       + 1/ϵr_N/rλ - λ^*_2(1, X, S) - (1, X, S)_2.

Here 

    r_N/rλ - λ^*_2 = r_N/r - 1λ + λ -  λ^*_2.


We can decompose the estimation error of ξ̂_1^rev
 as follows:
 
    ξ̂_1^rev - ξ_1^* 
           = 
        1/K∑_k=1^K [ [^N_kψ_1(W; r̂_N, η̂̃̂_k) - _Nψ_1(W; r̂_N, η̂̃̂_k)|η̂̃̂_k;     - ^N_kψ_1(W;  r_N, η̃^*_k) - _N
    ψ_1(W;  r_N, η̃^*_k)] ]_ℛ_7
       + 1/K∑_k=1^K _Nψ_1(W; r_0, η̂̃̂_k) - ψ_1(W; r_N, η̃^*)_ℛ_8 + 1/N∑_i=1^N ψ_1(W_i, N; r_N, η̃^*)_ℛ_9.

Here ℛ_7 is a stochastic equi-continuity term. It is o_p(^-1/2) because of the sample splitting and r_N - r = o_p(1) and η̂̃̂_k - η̃^*_2 → 0. 
The term  ℛ_8 can be bounded as follows: 

    ℛ_8≤1/ϵê_k - μ̂_k(1, X) - (1, X)_2 + Cr_N/r̂_N - 1 + λ̂_k - λ^*(1, X, S) - _k(1, X, S)_2

Under the assumptions in <Ref>, we know that ℛ_8 = o_p(^-1/2).

Therefore, we have 

    √()   ξ̂_1^rev - ξ_1^* 
       = √()/N∑_i=1^N ψ_1(W_i, N; r_N, η̃^*) + o_p(1) 
       = r_N^1/2/N^1/2∑_i  = 1^N {R_i, N/r_NT_i, Nλ^*(X_i, N)/(X_i, N)(Y_i, N - (T_i, N, X_i, N, S_i, N))}
       +r_N^1/2/N^1/2∑_i = 1^N{T_i, N/(X_i, N)((1, X_i, N, S_i, N) - (1 , X_i, N)) } +  r_N^1/2/N^1/2∑_i = 1^N {(1 , X_i, N) - ξ_1^*} + o_p(1)






We denote S_N = ∑_i = 1^N Z_i, N where 

    Z_i, N 
        = r_N^1/2/N^1/2[R_i, N/r_NT_i, Nλ^*(X_i, N)/(X_i, N)(Y_i, N - (T_i, N, X_i, N, S_i, N)) 
           + T_i, N/(X_i, N)((1, X_i, N, S_i, N) - (1 , X_i, N)) 
       + (1 , X_i, N) - ξ_1^*].


It is easy to verify that _N[S_N] = N_N[Z_i, N] = 0. Moreover, 

    _N[S_N] 
           = r_N_N[R_i, N/r_NT_i, Nλ^*(X_i, N)/(X_i, N)(Y_i, N - (T_i, N, X_i, N, S_i, N))]^2 
       + r_N_N[T_i, N/(X_i, N)((1, X_i, N, S_i, N) - (1 , X_i, N))]^2 
       +  r_N_N[T_i, N/(X_i, N)((1, X_i, N, S_i, N) - (1 , X_i, N))]^2  
       →{[Tλ^*(X)/(X)(Y - (T, X, S)) ]^2| R = 1}
       +  (R = 1)[T/(X)((1, X, S) - (1 , X))]^2 + (R = 1)[T/(X)((1, X, S) - (1 , X))]^2.


To verify the  Lyapunov condition, we note that for q > 2, {[S_N]}^q = O(1), and 

    _N[∑_i = 1^N |Z_i, N|^q] 
       = N_N[|Z_i, N|^q] = NZ_i, N^q_q 
       ≤r_N^q/2/N^q - 2/2[R_i, N/r_NT_i, Nλ^*(X_i, N)/(X_i, N)(Y_i, N - (T_i, N, X_i, N, S_i, N)) _q  
       + T_i, N/(X_i, N)((1, X_i, N, S_i, N) - (1 , X_i, N))_q + (1 , X_i, N) - ξ_1^*_q]^q

Note that 

    R_i, N/r_NT_i, Nλ^*(X_i, N)/(X_i, N)(Y_i, N - (T_i, N, X_i, N, S_i, N)) _q 
       = r_N^-q-1/q{_N[R_i, N/r_NT^q_i, Nλ^*q(X_i, N)/^q(X_i, N)(Y_i, N - (T_i, N, X_i, N, S_i, N))^q]}^1/q
       = r_N^-q-1/q{[T^qλ^*q(X)/^q(X)(Y - (T, X, S))^q| R = 1]}^1/q
       ≤C/ r_N^q-1/qϵ{[(Y - (T, X, S))^q| R = 1]}^1/q≤C^2/ r_N^q-1/qϵ, 
       T_i, N/(X_i, N)((1, X_i, N, S_i, N) - (1 , X_i, N))_q 
       ≤1/ϵ(1, X_i, N, S_i, N) - (1 , X_i, N)_q ≤C/ϵ
       (1 , X_i, N) - ξ_1^*_q 
        ≤ C.

Therefore, 

    _N[∑_i = 1^N |Z_i, N|^q] ≤1/(Nr_N)^q - 2/2C^2/ϵ + r_N^q/2/N^q - 2/2C/ϵ  + r_N^q/2/N^q - 2/2C → 0.

By Lindberg-Feller Central Limit Theorem, we have 

    √()ξ̂_1^rev - ξ_1^*  = S_N   d→ N(0, Ṽ_1),

where 

    Ṽ_1
               = {[Tλ^*(X)/(X)(Y - (T, X, S)) ]^2| R = 1}
       + (R = 1)[(1, X) - ξ^*_1]^2  + (R = 1)[T/(X)((1, X, S) - (1 , X))]^2.

Therefore, 

    _l^1/2(ξ̂_1^rev  - ξ^*_1) d→ N(0, Ṽ_1).

We can similarly prove that the asymptotic distribution for δ̂^rev is 

    _l^1/2(δ̂^rev  - δ^*) d→ N(0, Ṽ).


 Obviously when (R = 1) = 0, Ṽ equals Ṽ^* in <Ref>.

We can straightforwardly extend the proof of <Ref> to the current triangular array asymptotics and show the same result, ,  

    √(N)(δ̂ - δ^*) d→𝒩(0, V^*),

where 

    V^* = [ψ^2(W; δ^*, η^*) ],

with 

    [ψ^2(W; δ^*, η^*)] 
           = [(1, X) - (0, X) - δ^*]^2 
       + [T/(X)((1, X, S) - (1 , X)) - 1 - T/1 - (X)((0, X, S) - (0 , X))]^2 
       + [ TR/(X)(X)(Y - (1, X, S)) - (1 - T)R/(1 - (X))(X)(Y - (0, X, S))]^2.

By Slutsky's lemma and _l/N = r_N →(R = 1),

    √(_l)(δ̂ - δ^*) =  √(_l/N)√(N)(δ̂ - δ^*)d→𝒩(0, (R = 1)V^*).

The conclusion then follows from the fact that 

    (R = 1)[ TR/(X)(X)(Y - (1, X, S)) - (1 - T)R/(1 - (X))(X)(Y - (0, X, S))]^2 
    
     =   ^2(R = 1){[ T/(X)(X)(Y - (1, X, S)) - (1 - T)/(1 - (X))(X)(Y - (0, X, S))]^2 | R = 1}
    
     =   {λ^*^2(X)[ T/(X)(Y - (1, X, S)) - (1 - T)/(1 - (X))(Y - (0, X, S))]^2 | R = 1}
    
     =   [ψ̃^2(W; δ^*, η̃^*) | R = 1]

where the second equality follows from the fact that λ^*(X) = (R = 1)/(X).





 §.§ Proofs for <Ref>


In order to prove that T ⊥ Y | X, S, we need to verify that for any x ∈𝒳, s ∈𝒮, and y ∈𝒴,

    (Y ≤ y | T = 1, X = x, S = s) = (Y ≤ y | T = 0, X = x, S = s),

or equivalently, 

    (Y(1, s) ≤ y | T = 1, X = x, S(1) = s) = (Y(0, s) ≤ y | T = 0, X = x, S(0) = s).


We note that condition <ref> in <Ref> implies that 

    (Y(0, s) ≤ y | T = 0, X = x, S(0) = s)  = (Y(0, s) ≤ y | T = 0, X = x), 
    (Y(1, s) ≤ y | T = 1, X = x, S(1) = s)  = (Y(1, s) ≤ y | T = 1, X = x).

Then the condition <ref> in <Ref> implies that 

    (Y(0, s) ≤ y | T = 0, X = x) = (Y(1, s) ≤ y | T = 0, X = x).

Moreover, the condition <ref> in  <Ref> implies that 

    (Y(1, s) ≤ y | T = 0, X = x) =  (Y(1, s) ≤ y | T = 1, X = x).

These equations together ensure <ref>. 




 §.§ Proofs for <Ref>


Note that under assumptions in <Ref>,

    ξ_1^* 
           = [Y(1)] =  [[Y(1) | X]] =  [[Y | X, T = 1, R = 1]] 
       =∬ yf^*_X(x)f^*_Y | X, T = 1, R = 1(y | x, T = 1, R = 1)dxdy.

According to the assumption, we know the distribution of (T, X), thus we know (X) = (T = 1 | X) and f^*(X). The only unknown component that also contributes to the definition of ξ_1^* is the distribution of Y given X, T = 1, R = 1. Therefore, we can equivalently consider the efficiency lower bound for ξ^*_1 under the following model that fix all other components but the distribution of Y given X, T = 1, R = 1 at the truth.

    ℳ_fix   = {f_X, T, Y | R = 1(X, T, Y) = f^*_X | R = 1(X)[(1, X)^T(1 - (1, X))^1 - T]f_Y | R = 1, T, X(Y, T, X): 
                                      f_Y | R = 1, T, Xis an arbitrary density function}
 
It is easy to verify that the tangent space of ℳ_fix is Λ with orthogonal complement Λ^⊥_fix given as follows:

    Λ = Y | X, T, R∈ L_2(Y, X, T, R): Y | X, T, R| R = 1, T, X = 0
       Λ^⊥_fix = {X, T, S∈ℒ_2(X, T): [X, T| R = 1] = 0}


Again we consider parametric submodels indexed by γ in path-differentiability analysis for ξ^*_1. In the following analysis, we suppress the subscripts in the density functions to ease the notations. 

    ∂/∂γ_γ[Y(1)] |_γ = γ^*
       =  ∫ f^*(x) ∂/∂γ_γ[Y | X = x, T = 1, R = 1] |_γ = 0dxds 
       = ∫ f^*(x)[Y ×Y | X, T, R| X = x, T = 1, R = 1]dx  
       = ∫ f^*(x)T/(X)(Y - (T, X))Y | X, T, R| X = x, R = 1dx 
       = f^*(X)/f^*(X | R = 1)T/(X)(Y - (T, X))Y | X, T, R| R = 1

This means that ψ̃_I, 1(W; δ^*, η̃^*) = f^*(X)/f^*(X | R = 1)T/(X)(Y - (T, X)) is an influence function for ξ^*_1. Moreover, we can easily verify that  ψ̃_I, 1(W; δ^*, η̃^*) is orthogonal to Λ^⊥_fix. Therefore, ψ̃_I, 1(W; δ^*, η̃^*) is the efficient influence function for ξ^*_1. Analogously, we can prove that the efficient influence function for δ^* is ψ̃_I(W; δ^*, η̃^*) in setting I. 

Moreover, following the proof of <Ref>, we can analogously prove that the efficient influence function for δ^* in setting II is the same, ,  ψ̃_II(W; δ^*, η̃^*) = ψ̃_I(W; δ^*, η̃^*).

Therefore, the efficiency lower bounds for these two settings are 

    Ṽ^*_I   = Ṽ^*_II = [ψ̃^2_I(W; δ^*, η̃^*) | R =1 ]  
       = [[ Tλ^*(X)/(X)(Y - (1, X)) - (1 - T)λ^*(X)/1 - (X)(Y - (0, X))]^2| R =1 ] 
       = [λ^*2(X)/(X)(Y^2(1) - ^2(1, X)) + λ^*2(X)/1 - (X)(Y^2(0) - ^2(0, X)) | R = 1].

Note that the efficiency lower bound in <Ref> is 

    Ṽ^* 
           = [ψ̃^2(W; δ^*, η̃^*) | R =1 ] 
       = [[ Tλ^*(X)/(X)(Y - (1, X, S)) - (1 - T)λ^*(X)/1 - (X)(Y - (0, X, S))]^2| R =1 ] 
       = [λ^*2(X)/(X)(Y^2(1) - μ̃^*2(1, X, S)) + λ^*2(X)/1 - (X)(Y^2(0) - μ̃^*2(0, X, S)) | R = 1].

Therefore, the efficiency improvement from surrogates is 

    Ṽ^*_I  - Ṽ^* 
               = Ṽ^*_II - Ṽ^* = [λ^*2(X)/(X)(μ̃^*2(1, X, S) - μ^*2(1, X)) + λ^*2(X)/1 - (X)(μ̃^*2(0, X, S) - μ̃^*2(0, X)) | R = 1] 
       = [λ^*2(X)/(X){(1, X, S(1))| X} + λ^*2(X)/1 - (X){(0, X, S(0))| X}| R = 1].





The efficiency bound V^*_i directly follows from the efficiency bound V^*_IV in <Ref>. The efficiency bound V^*_ii directly follows from the efficiency bound V^*_III in <Ref> and the reformulation in <Ref> <ref>. The efficiency lower bound V^*_iii directly follows from the efficiency bound Ṽ^* in <Ref>.  The formula for efficiency gaps V^*_i - V^*_ii and V^*_ii - V^*_iii follow from the algebraic facts below:

    V^*_i  
           = [1/(X)(Y^2(1)- ^2(1, X)) + 1/1 - (X)(Y^2(0)- ^2(0, X))]^2 + [(1, X) - (0, X) - δ^*]^2, 
    
    V^*_ii   = [1/(X)(Y^2(1)- μ̃^*2(1, X, S(1))) + 1/1 - (X)(Y^2(0)- μ̃^*2(0, X, S(0)))]^2 
       + (R = 1)[1/(X)(μ̃^*2(1, X, S(1)) - ^2(1, X)) + 1/1 - (X)(μ̃^*2(0, X, S(0)) - ^2(0, X))]^2 
       + (R = 1)[(1, X) - (0, X) - δ^*]^2, 
    
    V^*_iii   = [1/(X)(Y^2(1)- μ̃^*2(1, X, S(1))) + 1/1 - (X)(Y^2(0)- μ̃^*2(0, X, S(0)))]^2.






§ ADDITIONAL NUMERICAL RESULTS









