PUATE: Efficient ATE Estimation from Treated (Positive) and Unlabeled Units
Authors: Masahiro Kato
Fumiaki Kozai
Ryo Inokuchi
Venue: NeurIPS 2025 poster ()
Keywords: Causal inference
PU learning
average treatment effect estimation
Primary area: probabilistic_methods

Abstract: The estimation of average treatment effects (ATEs), defined as the difference in expected outcomes between treatment and control groups, is a central topic in causal inference. This study develops semiparametric efficient estimators for ATE in a setting where only a treatment group and an unlabeled group—consisting of units whose treatment status is unknown—are observed. This scenario constitutes a variant of learning from positive and unlabeled data (PU learning) and can be viewed as a special case of ATE estimation with missing data. For this setting, we derive the semiparametric efficiency bounds, which characterize the lowest achievable asymptotic variance for regular estimators. We then construct semiparametric efficient ATE estimators that attain these bounds. Our results contribute to the literature on causal inference with missing data and weakly supervised learning.

PUATE: Efficient ATE Estimation from Treated (Positive) and Unlabeled Units
Masahiro Kato
Fumiaki Kozai
Ryo Inokuchi
Mizuho-DL Financial Technology Co., Ltd.
Chiyoda-ku, Tokyo 102-0083
masahiro-kato@fineee ienettoffinee...mm co.
Abstract
The estimation of average trealment effects (ATEs), defined as the difference in expected outcomes between treatment and control groups, a central topic causal inference. This study develops semiparametrie efficient estimators for ATE in a setting where only treatment group and an unlabeled group consisting units whose treatment status is unknown a variant of learning from positive and unlabeled data (PU learning) and can be viewed special case of ATE estimation with missing data. For this setting, we derive the semiparametric efficiency bounds, which characterize the lowest achiev- able asymptotic variance for regular estimators. We then construct semiparametric efficient ATE estimators that attain these bounds. Our results contribute to the lit- erature causal inference with missing data and weakly supervised learning.
observed. This scenario constitutes
Introduction
The estimation of average treatment effects (ATEs), defined as the difference in expected out- comes between treatment and control groups, is fundamental problem in causal inference (Imbens & Rubin, 2015).. Estimating ATES enables researchers to quantify the causal impact of treatment, intervention, or policy
outcome of interest. This problem is of paramount impor- tance across various fields, including economics, epidemiology, and machine learning.
Standard ATE estimation typically assumes access to both treatment and control groups, along with complete information on treatment assignment. However, in many practical situations, this assump- tion does not hold. In some cases, only a treatment group and an unknown group troup-comprisngg comprising units for which treatment assignment unobserved-are available. Such scenarios arise in various applications, including recommendation systems with implicit feedback, electronic health records, and marketing campaigns, where the absence of explicit treatment information poses significant challenges for causal inference.
We present the following examples for applications of our proposed method:
We are interested in how building online store affects product sales. In recommen- dation system, a customer who purchases product through the website known to have visited the site, whereas customer who buys the product in physical store may may not have visited the website. When the construction of the online store regarded as the treatment, eustomers who purchase in-store belong the unknown group.
In similar vein, consider the effect of distributing coupons on product sales. Because the logging system is imperfect, we observe two datasets: purchase records for customers who used coupons, and purchase records that mix customers who did and did not use coupons. Here, customers who appear only in the mixed dataset constitute the unknown group.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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A company unintentionally sold defective products, for example, vehicles with faults or food containing impurities. Consumers who notice the defect report both the defect and any resulting damage, such as health issues or accidents, to the company. Consumers who do not notice the defect file no report. In this setting, consumers who report the defect form the treatment group, while those who do not notice the defect belong to the unknown group.
Our method also useful when the treatment can be defined only by contrasting it with non-treatment condition. For example, consider examining how excessive work hours affect work- ers' mental health. Courts and labor inspectors can identify illegal overtime, but difficult to decide whether other working hours are excessive. Firms found to have engaged in illegal overtime can therefore serve as the treatment group. whereas all other firms constitute the unknown group. Similarly, by taking units with easily observable outcomes the treatment group and all others as the unknown group, one can define the comparison group simply as not treated. This idea is widely adopted in anomaly-dett -eetcctntttttion approaches that rely on density-ratic PU learning.
Content of this study. We address the problem of ATE estimation using only a treatment group and an unknown group. This setting is closely related to learning from positive and unlabeled data (PU learning, Sugiyama et al., 2022), where the goal is train a classifier using only positive and unlabeled instances. In our context, the challenge lies in efficiently estimating ATES using the treatment (positive) and unknown groups. We refer to our setting and methodology as PUATE.
For this problem, we first derive semiparametric efficiency bounds, which are theoretical lower bounds the asymptotic variance of regular estimators under the given data-generating processes (DGPs) These bounds serve as benchmarks for evaluating estimator performance. As part of this derivation, we compute the efficient influence fianctio, which provides insight into the construction of efficient estimators.
Using the efficient influence function, develop semiparametric efficient ATE estimators that yn-consistent and whose asymptotic variance achieves the efficiency bounds. These estimators are thus optimal under the semiparametric framework.
In this study, we consider two DGPs relevant to the PU setup: the censoring setting and the case- control setting (Elkan & Notd, 2008; du Plessis et all 2015). In the censoring setting, we are given a single dataset in which some units have missing treatment information while others are confirmed provided with two datasets: one
to have received treatment. In the case-control setting, containing treated units and another comprising units with unknown treatment status.
Specifically, our contributions are as follows:
We formulate the ATE estimation problem with missing data using the PU learning framework. We derive efficiency bounds under both the censoring and case-control settings.
We propose novel efficient estimators.
We establish connections between ATE estimation with missing data and PU learning.
We also propose alternative candidate estimators.
This study is organized as follows. Scction 2 formulates our problem. Section introduces simple ATE estimators, which are later shown to be inefficient. In Section derive efficiency bounds, propose an efficient estimator, and establish its asymptotic properties under the censoring setting. Due to space limitations, we show the ATE estimation the case-control setting briefly in Section S and mainly in Appendix D Section 6 presents simulation studies. We introduce related work in Appendix[A] A The details of PU learning methods are explained in Appendix
2 Problem setting
2.1
Potential outcomes and parameter of interest
We consider binary treatments, and O. For each treatment d E {1,0). there exists potential outcome (d) R. The outcome observed only when the corresponding treatment is assigned
For regular estimators, p.366 in/van der Vaart (198).

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to unit. Each unit has p-dimensional covariates X X' C RP. This setting is called the Neyman- Rubin causal model (Neyman. 1923: Rubin, 1974).
We denote by Po E P a distribution of Y(d), X, and other random variables introduced below, which we call the true distribution, where P is the set of distributions. For simplicity, we assume that Po has a density. We denote the conditional density of Y (d) given X by pr(a),(3(()) X), and the marginal density of X by (u(x))
We consider units, each of which receives either treatment or control. Let Yi(d) and X; be i.i.d. copies of (d) and X. Throughout this study, for a random variable R. R2 its i.i.d. copy under Po. If unit i receives treatment d, we observe (d) but not the counterfactual outcome.
We refer the group of units who receive treatment 1 as the treatment group and those who receive treatment 0 as the control group. As formulated in the next subsection, we consider a scenario where part of the treatment group and a mixture of the treatment and control groups are observable, where the treatment indicator is unobservable. We refer to this mixed group as the unknown group.
Our objective is to estimate the ATE under Po using observed data, defined as To E (1 - Y (0)), where E denotes the expectation under Pn-
2.2 Observations with two DGPs
In our setting, the observations are non-standard. We can only observe part of the treatment group and the unknown group, a mixture of the treatment and control groups. This setting a variant of PU learning. PU learning encompasses two settings: the censoring setting and the case-control setting. In the censoring setting, we consider a single dataset with i.i.d. observations, where treatment labels contain missing values. Specifically, while part of the treatment group is observed, the mixture of the treated and control groups is also present. In the case-come case-control setting, we observe two independent datasets: one consisting solely of the treatment group and the other comprising the unknown group. The censoring and case-control settings are also referred to as one-sample and swo-sample settings, respectively (Niu et all 2016). The case-control setting can also be regarded as form of stratified sampling, studied in Imbens & Lancaster (199) and Wooldridge (2001).
2.3 Censoring setting
In the censoring setting, we observe single dataset D, defined as follows:
D Ox, with (X, Y.) po(x,,,9),
where O, (1,0} is observation indicator with the observation probability #0 X), Y, is defined as
Y 1[0, 1])x + 1[0, -o]
Y is the observation of the unknown groups and defined as
Y = 1(: 1] + 1[D, (0),
D, E [1,0) is a (unobserved) treatment indicator, and we denote the conditional probability of D; given X and O = 0 as go X)= = TP(D X,O 0). We refer to go X) the censoring propensity score. Here, the density Po y) is given as po o, y) = oo / X = x)Py:o(U 0 X x), where PY, is the density of Y.
2.4 Case-control setting
In the case case-control setting, we observe two stratified datasets, Dr and Du:
Dr == (XTr,, with (Xr,,, ())) ~ (ro((a,(())) and
Du (Xx,, (Xx,Yu,)) Yu, with (Xx, Yu,k) pu,o yu),
where m and are fixed sample sizes of each dataset such that m +1 = n, Xr, represents the covariates of the treatment group, Yu, the observed outcome defined as
Yu, 1,2 (1) + 1[D (0),
3

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and D2 E {1.0} is a treatment indicator with probability co(D | X). We refer to cold X = x) as the case-control propensity score. The densities (T, y(1) and pt.o(z, Mt) sat- isfy (r,o(1,,(())
X x) respectively, where PYc,o oyy X x) denotes the density of YU,& given X and (T,O represents the density of the covariates in the treatment group.
= (r.0(2))y())00((
| = x) and pu,o(1, yv) = (o(z)py,
Although the ATE estimation in the case-control setting is also crucially important, due to the space limitation, show the main results almost in Appendix D
2.5 Difference between the two settings
We illustrate the concept of the censoring and case-control settings in Figure [ A summary of the differences is provided below:
Pepultion all bedtene inboaar (K.1)
ATE defeed
Ransian sampling
Trasteett goup
Treatnent grop
Censoring setting: A single dataset is ob-
served, containing partial treatment informa- tion and mixture of treated and control groups.
(2:13
Uninowe PWA
Case-control setting: Two stratified datasets are observed one consisting of the treat- ment group and the other comprising the un- known group.
Figure Illustration of the censoring and case- control settings
The key distinction lies in the randomness of treatment group observations. In the censoring setting, the observation of the treatment group is random event, where the observation indicator O, follows a probability (O X). In contrast, in the case-control setting, the label observation is determin- istic, and the treatment and unknown groups are drawn independently. This difference impacts the estimator design and efficiency bounds.
Note that the definitions and meanings of the propensity scores in the censoring and case-control settings are also different, and its difference stems from the definition of the observations indicators.
Notation. We summarize the notations above and introduce notations. Let E, P. and Var be an expectation operator, a probability law, and a variance operator. For both settings, let us define (X) = JE (1) X). (X)) = E[Y(O) (0) | X], and let (x(X) pr.,((X) (X,)(X) be the conditional ATE. If a function f depends on the true distribution Po. we denote it by fo-
In the censoring setting, we use Tolo X) = P(O
X), (o X) = P(D = d X, X = 0), X) ()x),((X) and E[
and vn (X) Blÿ X. 0 0). Here, E[ O 1]Y X] = To
o X] To X Po (X) (0 X) X X) (X) + (0 X (0 X Pc.00 () hold
under Assumption
defined later.
In the case-control setting, we use eg(d X) = P(D = d X) and vu,o(X) = Y X] Here, (X))(X) co X))TT,(() + co ())c,o(X) holds under Assumption C,3
Let
rx( = (x(X) be the density ratio between the covariate densities.
(r,((X)
Example of ATE estimators in the censoring setting
In this section, preliminary, suggest simple ATE estimator in the censoring setting. In Appendix [D] we show the detailed result of the estimator and also propose a simple ATE estimator in the case-control setting. Note that as discussed in the subsequent subsections, the estimators not efficient, i.., there exist ATE estimators whose asymptotic variance is smaller.
3.1 The Inverse probability weighting estimator
We first consider the censoring setting. We begin by the arguments from the estimation of the propen- sity score. To estimate the propensity score, we employ the result in PU learning. Elkan & Noto (2008) addresses PU learning in the censoring setting. In that work, they show that under the Se- lected Completely At Random (SCAR) assumption defined below (Elkan & Noto, 2008). This as- sumption is analogy of the Missing Completely At Random assumption (MCAR), which is common

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with the missing data literature (Little & Rubin, 2002: Rubin. 1974, Bekker & Davis, 2020). Sev- eral works such as Bekker & Davis (2018) attempt relax the SCAR assumption, but to relax the assumption, we usually require additional assumptions.
Assumption 3.1 (SCAR). holds that P = 1,0 - o xx) P(D
X)P(( - D= 1).
From the assumption, we have To(1 x) PP
x)P(O D 1) since P(D
0, - 1 X)
holds by definition of the DGP. Thus, under this assumption, the propensity scores 9u (d ax) and fd x)) can be estimated using PU learning methods under some conditions ddu Plessis et al. 2015: Elkan & Noto, 2008 we know their true values, such assumptions are unnecessary. Note that the censoring PU learning studies aim to estimate P(D d X) not go(d X) = = = d X = 0). However, once we obtain an estimate of IP ( = 1 X) and To X),
can obtain go X) from g1 X) PPO D 1)P(D 1 X) /T/To((0 X).
We further make the following unconfoundedness and common support assumptions, which are common in ATE estimation.
Assumption 3.2 (Unconfoundedness in the censoring setting). The potential outcomes (Y(1), independent of treatment assignment given covariates: (1),,, (L)) D) X.
Assumption 3.3 (Common support in the censoring setting). There exists a constant e independent of such that for all X, To(o 22), go({ x), Co(x) hold.
Under these assumptions, the ATE is estimable by replacing the following two quanti- ties with sample approximation: ([ = E 10-1jy [ TE [ and [[(( =
(o, where go and To can be estimated, and expectations can be approximated by sample averages. Such an estimator is
(o (o)] x0([1)1[0=111 = E _o
10=0
(X) xx1l
variant of the inverse probability weighting (IPW) estimator and shown in Remark
3.2 Toward eflicient estimators
Thus, we can estimate the ATE in the censoring setting (and in the case-control setting, shown in Section and Appendix C.2). However, it is unclear whether it is efficient; that is, the (asymptotic) variance is sufficiently small. In the subsequent subsections, we investigate efficient estimators and develop efficiency bounds, which work as lower bound for the regular estimators. The efficiency bound also suggest the construction of efficient estimators, and actually propose estimator whose asymptotic variance aligns with the efficiency bound.
Semiparametric efficient ATE estimation under the censoring setting
This section presents method for ATE estimation under the censoring setting. First, derive the efficiency bound in Section [. Then, we propose our estimator in Section [4. and show the consistency in Section 4.3 and the asymptotic normality in Section 4.4 Finally, in Section 4.5 discuss issues related to the estimation of the propensity score.
4.1 Efficient influence function and efficiency
First, we derive the efficiency bound for regular estimators, which provides a lower bound on asymptotic variances. The efficiency bound characterized via the efficient influence function (van der Vaart [1988,, derived as follows (Proof is provided in Appendix H):
Assumption 4.1 (Regularity conditions). There exists a constant C > 0 independent of such that vo (T, E C-C,,Cl.
The outcome Y and covariates X have finite variances.
'ddentifiability of the propensity scores differs between the censoring and case-control settings, and in both cases, it further depends on specific assumptions. [Ekan & Notd (2008) shows that, under the censoring setting, the conditional class probability can be learned without the class prior, provided that the SCAR assumption holds. In the case-control setting, existing studies often require the class prior in advance of the conditional probability estimation, Some studies demonstrate that both the propensity score identified simultaneously if parametric models are used for the propensity score (Lancaster & Imbens, 1996. Kato et all, (2018). There are also studies that investigate class prior estimation as an independent problem RRamaswamy et all 2016. du Plessis & Sugivama, 2014).
class prior can be
5

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Lemma 4.2. If Assumptions 3.2 3.3
4.1 hold, then the efficient inftuence function is given as
Trens (X, O, : Vo, To. 9o To), where
srcens (X, O, Y :PT, No, To. 90. TO) = Seens (X, NK,,O,iHee W.T;H Vo, TO, To)
1 100 - 1| (xT,o())
1 0) v0
grens (X, S, Y; PT,,0, Po: No. 9o)
To X)
90 X)=o(0 X)
+ go(1 X)][0 1 110----- -x,,(()))
go(1 XX)
+ (T,,(()) 90 x) vo (X
9o (0 x) (T,,(XX)
90 TT X)
Here, note that the efficient influence function depends on unknown JMT,0: Mo, no 9o. which are referred to as nuisance parumeters. Since the efficient influence function satisfies the PT,,0 ; Mo, To, 9o,9o ro]] and the exact expectation is computed, we can obtain To by solving for To that satisfies O. Thus, the efficient influence function provides sig-
equation E [Tpees (,
0, if the nuisance parameters are known
E <[yeeeu(X, (X, O, nificant insights for constructing an efficient estimator. Furthermore, the accuracy of the estimation of the nuisance parameters affects the estimation of To. the parameter of interest.
PT.,0, HT..O Vo, To. 9o. To)]
From Theorem 25.20 in van der Vaart 1998). Lemma yields the following result about the effi- ciency bound.
Theorem 4.3 (Efficiency bound in the censoring setting). If Assumpttonss asymptotic variance of any regular estimator is lower bounded by
3.2
hold, then the
Vemes
== E [ cponns (x, O,YY; Po, Vo, To. 9o. ]o
= E (1- (1 x)
2 Var (1)) x). + Var Y X)
+ (Tu(X)-
(0 xx)
To X)
(o ( X)
We say that an estimator is efficient if its asymptotic variance aligns with Vrees
4.2 Semiparametric efficient estimator
Based on the efficient influence function, we propose an ATE estimator defined Pn prens-eff "E= Scons (X,, O,, Yú; J i Dn,i, Rn,i, In,s), where Tr,,, Pn,i Rn,s and Zn.i are estimators of μr.o, Mu, To. and gu- Note that the estimators can depend i. This estimator is extension of the augmented inverse probability weighting estimator, also called a doubly robust estimator (Bang& & Robins, 2005).
Remark (Estimation equation). There exist several intuitive explanations for Tn Prons-off One of the typical explanations the one from the viewpoint of the estimation equation. Given NT.nn. Un,ū Rn.i and In.i, the estimator Ty feeus-ell is obtained by solving the following equation: inns,zoos-eff
"'= (X,,, Os, Y; (T Va,s, Rn,i In sgens-eff) = 0. Such derivation of the efficient estimator as the estimation equation approach as explained Schuler van der Laan 2024).
4.3 Consistency and double robustness
First, we prove the consistency result; that is, 7 pcens-eff Pg To holds as -> co. We can obtain this result relatively easily compared to the asympiotic normality. We make the following assumption that holds for most estimators of the nuisance parameters.
Assumption 4.4. There exist constants C1, C2 O independent of n such that (d X), Fn,s X) (C1, 1 C1) and (X), (r,,,,(X) [-C2, C2) holds almost surely. As - oc, || soll2 = (1) holds. Additionally, either of the followings holds for all 21.
| - no = (1).
|P.. - v voll2 = (1) and || - NT. = (1).
For the estimation of the censoring propensity score, we can employ the existing PU learning meth- ods in the censoring setting, such as Elkan & Noto (2008). Note that we can also apply methods for the case-control PU learning, such as du Plessis et al (2015), since, as the classification prob- lem, the case-control setting is more general than the censoring setting (Niu et al., 2016). Note that the case-control PU learning methods typically require the class prior P(D) = 1), which can be

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Algorithm 1 Cross-fitting in the censoring setting
Input: Observations D == {X,, Oz. J number of folds L, and estimation methods for JAT o. Mo, To- Let = 11.2...., n be the index set. Randomly split I into L roughly equal-sized folds, (eEe) Note that ULEc I(0) = I.
for E L do
Set the training data as I(-() (1,
...,,,,,, n) \ IT())
(00 (10) Construct estimators of nuisance parameters on I(-)), denoted by PT. Un =l0) end for Output: Obtain an ATE estimate T2
(1) Poens-eff using PT, Dn and (!!)
estimated under several additional assumptions, even if we do not know it (du Plessis & Sugiyama 2014: Ramaswamy et al. 2016: Kato et al. 2018).
Then, the following consistency result holds. The proof directly follows from the one for Theo-
rem 4.8
Theorem 4.5 (Consistency in the censoring setting). If' A'sssmpttions 3.2 and 4. 4 hold, then
pcens-eff & To holds as n + oc.
Double robustness. There exists double-robustness structure such that given |l 90 21 = (1) if either || - Toll2 = (1) or | - voll2 = op(1) and | - pT,,l12 = (1(1) holds, then prens-off P, To holds. Here, note that we need to estimate the propensity score consistently nn to estimate the ATE and the double robustness holds between the estimators of the observation probability un and the expected outcomes fr.o and 1/1 This is because, in our setting, the treatment indicator is unobservable. Under this setting, identify the ATE, need to use the propensity score and cannot avoid its estimation. Also see Appendix L3
4.4 Asymptotic normality
Next, we prove the asymptotic normality. Unlike consistency, we need to make stronger assump- tion on the nuisance estimators, especially for the propensity score.
To establish the asymptotic normality or vfr-consistency of the estimator, it is necessary to control the complexity of the estimators of the nuisance parameter. One of the simplest approaches is to assume the Donsker condition, but it is well known that the Donsker condition does not hold in several cases, such as high- high-dimensional regression settings. In such cases, asymptotic normality can still be established through sample splitting, a technique in this field (Klaassen. 1987), which has been recently refined by Chernozhukov et (2018) as cross-fitting.
Cross-fitting is a variant of sample splitting (Chernozhukov et al. domly partition D into L >> folds (subsamples), and for each fold é E = (1,2,... ..., L). the nuisance parameters are estimated using all other folds. We estimate pT,0; Vo. To. assuming that the (l0) propensity score Do is known. Let us denote the estimators fold E C as PT, D(0) =() Let (() be the set of the sample index belonging fold
Cross-fitting.
2018 We ran-
Various estimation methods can be employed, including neural networks and Lasso, provided they satisfy the convergence rate conditions specified in Assumption 47 We later relax the assumption of a known propensity score. It is important to note that issues related to the propensity score estimation cannot be fully addressed even with cross-fitting. The pseudocode is in Algorithm[4.4
Asymptotic normality. We describe the results only for the case with cross-fitting, but similar results hold for the case when we assume the Donsker condition.
We make the following assumptions.
Assumption 4.6. The propensity score gu is known (Gr,t 9u). Assumption 4.7. For each & E L, as n + cc, the followings hold:
'In the standard setting, the double robustness holds between the estimators of the propensity score and the expected outcome.
7

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nool X) o 12 (p) {1,0}, |xr,o(X) - (X)1l = 0p(1), and
(oo(X) - V/ (x) 12 = (1)
((0) To(d| x) (d x) (1.00) pT. (x) 0p(n 1/2) for d {1,0).
To(O X) (0 1l0 - 1/ (x)112 Op(n' (12).
Then, we construct the estimator as Prens-eff = EiEc Eiezio Scens (X,, Y;: T.T 1/ =10 9o) and show the asymptotic normality holds as follows:
Theorem 4.8 (Asymptotic normality in the censoring setting). Consider the censoring setting. Sup- -(f)
pose that Assumptions 3.2
4. 4.6 hold; that is, In. go, and fT, = P Dn 1/2
(E) and n. ni are constructed via cross-fitting with certain convergence rales. Then, have
Vn (Foens-eff - -To 9 NO, Veeus)
The proof is provided in Appendix M The asymptotic variance of Feeus-eff matches the efficiency bound. Therefore, Theorem 4.8 also implies that the estimator 7n
Prons-eff is asymptotically efficient.
We discuss the other candidates of ATE estimators below.
Remark (Inefficiency of the Inverse Probability Weighting (IPW) estimator). We can define the
IPW estimator as Frens-IPW Compared to our proposed efficient estimator, this estimator does not use the conditional outcome estimators (Horvitz Thompson, 1952). go and no are known, this estimator is unbiased. How-
1[0.=0]Y, E 100== (1)x() (0. x.)....0)) + 01. = (,, (1Ix))
ever, it incurs a large asymptotic variance, given as VIPW = E [[((1-
9o(1
(0)x))
[o[1|X)
E["21X]
90(0|X]=mol0]x] Here, it holds that VIeW 2 Vaes, where the equality holds when (T,(O)) 0
and vo = O hold for all x. Thus, the IPW estimator is inefficient compared to Ty
qcens-eff, Addition-
ally. if nu unknown, the IPW estimator requires more restrictive conditions for the asymptotic normality than Frensvefft
Remark (Direct Method (DM) estimator). Another candidate is DM estimator, defined as -cens-Da DM == (X) - In. Pu, gn.al1|X)
To.x ), which is also referred as naive plug-in estimator. The asymptotic normality significantly depends on the estimators TTT, and In Additionally, the DM estimator known to sensitive to model misspecification.
In
4.5 Unknown propensity score
We have assumed that the propensity score gu known. This is because we cannot establish yna- consistency even if we assume the Donsker condition employ cross-fitte fotss-fitting if go is estimated. However, this assumption can be relaxed by utilizing an additional datasct to estimate Go-
Several practical scenarios exist. For instance, consider that the following additional dataset is available: Daux = )x) (Xe,Op) ~ So (o(z)To(o x))
Such dataset can be less costly since does not have the outcome data. Let @n be an estimator obtained from Daux and consider the following assumption:
Assumption 4.9. It holds that "u g l2 (1) nux
If aux approaches infinity independently of n, under Assumption 3.1 establish the asymp- totic normality without assuming the propensity score is known. Corollary 4.10 (Asymptotic normality in the censoring setting). Consider the censoring setting.
Suppose that Assumptions (0, ycens) as πi + oc.
and hold. Then, it holds that vn (poens-eff - Tp) $
We can also use {(X,, 0,) from D to estimate go with Daux The inclusion of 0,) can improve empirical performance.
Another practical scenario involves an auxiliary dataset with treatment indicators and missing out- comes, given as Daux' = {(xx, Dz) )r (Xv, Dr) (o(2)00(o x)).
8

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Table 1: Experimental results. Left: censoring setting; Right: case-control setting.
Censoring
IPW
DM
Eifictent
IPW
DM
Eficient
Case-
IPW
DM
Efficient
IPW
DM
Efficient
festimated go)
[true
96)
control
(estimated eq)
(true
en)
MSE
0.31
0,008
0.06
0.06
0.01
0.01
MSE
10.35
0.07
0.06
0.03
0.01
0.00
Bias
-0.26
0.16
0.12
-0.06
0.03
0.00
Bias
1.44
0.11
0.07
0.00
0.03
0.00
Cov. ratio
0.95
0.07
0.78
1.00
0.09
0.93
Cov. ratio
0.57
0.61
0.73
0.98
0.95
0.95
Estimates (censoring setting)
Estimates
setting)
IPW (ettmmate go)
(edhmated go
Enplical Gendoa
CM festimated
GM cestimaate gol
offcieet jestimtee
(true gt)
DM tiue
OM thrs ge!
Effiient true AlE
Mixient
2.00
2:25
2.50
2.75
300
325
3.50
375
4.00
22
24
2.5
2.0
3.0
32
34 36
3.
AEE
Figure 2: Empirical distributions of ATE estimates.
Remark (Double machine learning). Semiparametric efficient estimators typically rely on two in- gredients: doubly robust structure and control of function complexity. our setting, the doubly robust structure links the observation probability and the conditional expected outcome, but it does not involve the propensity score. Specifically, conditional on consistent estimator of the propen- sity score, if either the observation probability the conditional expected outcome is consistently estimated, the ATE can be estimated consistently. The doubly robust structure contributes not only to consistency but also to rapid bias reduction through Neyman orthogonality. In the standard ATE setup, one can remove bias by appropriately estimating the conditional expected outcome and the propensity score and plugging them into however, although a doubly robust relation holds between the observation probability and the con- ditional expected outcome, does not extend to the propensity score and the remaining nuisance parameters. Consequently, the estimation error of the propensity score may persist. Specifically, (n(F- G + Biasl Bias2, where G is asymptotically normal, Biasl is the product of the estimation errors for the observation probability and the conditional expected outcome, and Bias2 is the estimation error of the propensity score. Biasl can be Op(n -1/2), ), but Bias2 is not 0p 1/2) un- less the propensity score is estimated at a rate faster than the standard parametric rate Op(n 1/2). This is why our analysis assumes that the propensity score is known or is estimated more rapidly using an independent dataset. When using such an independent dataset, in our proof we take the limit with m - oo first, followed by n. + co. Alternatively, one can consider asymptotic regime which both nax and diverge (i.e., nu , - oo); in that case, m must diverge faster than n.
estimating equation. In PUATE,
5 Semiparametric efficient ATE estimation under the case-control setting
Here, we briefly introduce the ATE estimator in the case-control setting. More detailed results are shown in Appendix D
We define
poc-off
m E
(11x)) Ein (0)x))
(Y₂() (X)
( X.) Yu. ((.))) + (T,
Br. (Xx)
En (1 +
as an ATE es- timator in the case-control setting. Here, TT Mu,,n En.j. and Fr.j are estimators of μr, Mu.o- co, and ro. where m and I denote the dependence on each dataset.
en Xo)
en (1))))
For the estimator, we show the following theorem, which is an informal version of Theorem D.8
Theorem 5.1 (Asymptotic normality in the case-control setting (Informal)). Fix a E (0.1). For > O, consider the case-control setting with sample sizes m, 1 such that m an and | = (1 - a))n If the case-control propensity score en and the density ratio are known (En,; = eg and Pn,i = ro), and
-(E) TT..n 'T,m'' Fu,n, Puj are consistent estimators constructed via cross-fitting. Then, under
-(E)
regularity conditions (see Theorem D.&,, we have /ni (7n7790 (pec-elf TO) S No Vo0) as - co.
where y > O is the efficiency bound defined in Theorem D.3

---

Simulation studies
This section investigates the empirical performance of the proposed estimators. We also show the experimental results using semi-synthetic data in Appendix M
6.1 Censoring setting
We generate synthetic data under the censoring setting, where the covariates X are drawn from a multivariate normal distribution as X ~ <o(2;), where (o(x) is the density of (0, Ip), and Ip denotes the (p x p) identity matrix. We set = Set P(D X) trunc(sigmoid(X" ()). 3),0.1,0.9 where 3 is coefficient sampled from (O., 0.57p), and trunc(f, f, a. b) truncates ł by and b (a < b). Treatment D is sampled from the probability. The observation indicator O generated from Bernoulli distribution with probability if D; 1 and O = 0f D; O. Here, c generated from a uniform distribution with support [0, 1]. The outcome is generated as = X 3 To DEE where (0, 1), where we set 70 3.
The nuisance parameters are estimated using linear regression and (linear) logistic regression. We compared our proposed estimator, Tcons-eff with the other candidates, the IPW estimator Frens-IPW and the DM estimator a Prens.DM defined in Remarks 4.4 and 4.4 respectively. Note that all of these estimators are proposed by us, and our goal is not confirm Fr Accns-eff outperforms the others, while recommendation is Prens-off We consider both cases where the propensity score is either estimated using the method proposed by Elkan & Noto (2008) or assumed to be known.
We set 3000. We conduct 5000 trials and report the empirical mean squared errors (MSEs) and biases for the true ATE and the coverage ratio (Cov. ratio) computed from the confidence intervals in Table We also present the empirical distributions of the ATE estimates in Figure
As the theory suggests. coens-eff exhibits smaller MSEs compared to other methods. Interestingly, when the propensity score is estimated, the MSEs decrease, a phenomenon reported in existing studies. The coverage ratio is also accurate. The empirical distribution of the ATE estimates demon- strates the asymptotic normality.
6.2 Case-control setting
In the case-control setting, covariates for the treatment and unknown groups generated from different p-dimensional normal distributions: XT ~ (r,((z) and X ~ Co(x) = co(1)(,.(x)) + (z))) where set p 3, (ι,0(1) and (o(x) are the densities of normal distributions Npp Ip) and (Mp Ip). Mp
0.5 and pn, 0, 1p (1 the class prior set as co(1) = 0.3. By definition, the propensity score co(d x) is given as 60 x) 6o (T.
1)T and eo(1) is
The outcome is generated similarly to the censoring setting Y X 8 + 1.1 ToD +c, where To 3.
We set m 1000 and 2000 and compute the same evaluation metrics as in the censoring setting. Although logistic regression is used, the propensity score model is misspecified, while the expected conditional outcome follows a linear model.
Overall, pce-eff demonstrates robust performance in terms of MSE, bias, and coverage ratio. The poor performance of the IPW estimator is attributed to model misspecification.
We investigate non-linear settings in Appendix
7 Conclusion
In this study, we investigated PUATE, the problem of ATE estimation in the presence of missing treatment indicators. We formulated the problem using the censoring and case-control settings, in- spired by PU learning. For each setting, we derived the efficiency bound and developed an efficient estimator. Our analysis revealed that achieving asymptotic normality and efficiency. Future research directions include extending our approach to the semi-supervised setting, handling additional miss- ing values, and the relaxation of assumptions regarding the missingness mechanism.
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Contents
Introduction
2
Problem setting
2.1
Potential outcomes and parameter of interest
2.2
Observations with two DGPs
2.3
Censoring setting
2.4
Case-control setting
2.5
Difference between the two settings
Example of ATE estimators in the censoring setting
The Inverse probability weighting estimator
3.2
Toward efficient estimators
Semiparametric efficient ATE estimation under the censoring setting
4.1
Efficient influence function and efficiency
4.2
Semiparametric efficient estimator
4.3
Consistency double robustness
4.4
Asymptotic normality
4.5
Unknown propensity score
8
Semiparametric efficient ATE estimation under the case-control setting
Simulation studies
10
6.1
Censoring setting
10
6.2
Case-control setting
10
Conclusion
10
Related work
24
A.1
ATE estimation with missing data
26
A.2 Introduction of PU learning algorithms
26
Reformulation of the censoring and the case-control settings
27
I Reformulation of the censoring setting
27
B.2
Reformulation of the case-control setting
28
Details of the examples of ATE estimators censoring setting
28
Example ATE estimator in the censoring setting
28
C.2 Example ATE estimator in the case-control setting
29
Semiparametric efficient ATE estimation under the case-control setting
29
D.
Efficient influence function and efficiency bound
29
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D.2
Semiparametric efficient estimator
30
D.3 Consistency
30
Asymptotic normality
30
D.5
Comparison with the censoring setting
31
PU learning algorithms
Censoring PU learning
31
Case-control PU learning
................................
32
Density-ratio estimation
33
F
Remark on the nuisance parameter estimation in the censoring setting
33
Pseudo-code for ATE estimation in the case-control setting
33
Proof of Lemma
34
Proof of Theorem 4.8 Semiparametric efficient ATE estimator under the censoring setting
38
L.1
Proof of
40
1.2 Proof of 5
43
1.3 Remark on the double robustness
44
J Proof of Lemma
D.2
45
Proof of Theorem D.8: Semiparametric efficient ATE estimator under the case-control setting
48
Additional results of the simulation studies
49
Censoring setting
49
Case-control setting
50
M Empirical analysis using semi-synthetic data
52
M. Dataset
52
M.2 Censoring setting
52
M.3 Case-control setting
53
A Related work
The ATE estimation problem has long been studied in statistics, epidemiology, economics, and ma- chine learning (Imbens & Rubin, 2015). While randomized controlled trials are considered the gold standard, it is extremely important to estimate the ATE in observational studies. In ATE estimation with observational data, one of the basic approaches is employ the IPW estimator, which allows us to correct for selection bias Horvitz & Thompson 1952).
Although the IPW estimator is powerful tool, it is known that its asymptotic variance exceeds the efficiency bound even when the true propensity score is used (Hahn 1998)
*Under certain conditions, using an estimated propensity score can reduce the asymptotic variance, as shown by Hirano et al (2003). This phenomenon is known as the paradox of the propensity score (Henmi & Eguchi 2004 Kato al. 1).
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Another powerful estimator in this context is the doubly robust estimator (Bang & Robins. 20050, which also plays an important role in the literature on missing data (Yang et al. 2024). The dou- bly robust estimator not only satisfies the double robustness property but also achieves asymptotic efficiency; that is, its asymptotic variance reaches the efficiency bound (Hahn, 1998). This prop- erty is closely related to the efficient influence function in the derivation of the efficiency bound (van der Vaard [998; Tsiatis 2007). The doubly robust estimator is defined as a sample average of the efficient influence function with estimated nuisance parameters.
The doubly robust estimator has been refined in several directions. For example, yan der Laan (2006) propose the targeted maximum likelihood framework, which can improve the finite-sample performance of the efficient ATE estimator. Chernozhukov all (2018) propose debiased ma- chine learning framework, which is further generalized as automatic debiased machine learning
(Chernozhukov et all 2022) and Riesz regression (Chernozhukov et all (024)). Covariate balancing scores are also important attempts to improve performance (Imai & Ratkovic, 2013 Hainmueller 2012: ZZubizarretal 2015: Zhad 2019), and BBruns-Smith et all (2025) and Kato (2025a) show the
equivalence between covariate balancing methods and Riesz regression. In particular, Katd (2025a) and Kato (2026) propose a unified framework for debiased machine learning. Motivated by these works, Kato (2025h) proposes semi supervised framework for debiased machine learning that builds on arguments about the sampling scheme, as well as on ours.
To construct efficient estimators, convergence rate conditions and complexity restrictions for the nuisance estimators are usually required (Schuler & van der Laan, 2024).. In particular, to satisfy the complexity restrictions, researchers often assume the Donsker condition or apply sample split- ting (van der Vaart, 2002: Klaassen, 1987: Zheng & van der Laan, 2011). These approaches are further developed in the double machine learning framework by Chernozhukov et all (2018), where convergence rate conditions are relaxed through the use of Neyman orthogonality, and complexity restrictions are addressed via sample splitting, known cross-fitting For discussions of the relation- ship between double machine learning and other frameworks, such as targeted maximum likelihood estimation, see Kennedy (2016. 2023).
It is important to note that Neyman orthogonality and pross-fitting play different roles. Cross-fitting is used to ensure that the nuisance estimators (e.g., propensity score, outcome models) are indepen- dent of the observations to which they are applied. However, cross-fitting alone is not sufficient to guarantee asymptotic normality. The issue is that nuisance estimators typically converge at rates slower than vn. In doubly robust estimation, the convergence rate conditions for the nuisance estimators are relaxed due to the doubly robust structure, which is also referred to as Neyman or- thogonality.
Our work builds upon these arguments. However, in our setting, we cannot apply the techniques from Chernozhukov al. (2018) to mitigate the convergence rate condition for the propensity score. In other words, our estimator is sensitive to the accuracy of the propensity score estimation. There- fore, we assume that the propensity score is known in order to derive asymptotic normality, although consistency can still be achieved when the propensity score is estimated.
CATE estimation is also important topic related this study (Heckman et al. 1997). Var- ious methods have been proposed for estimating CATE including methods using neural net- works (Johansson et all 20016: [Shalit et all 2017: |Shi et all, 2201:: Hassanpour & Greinen 2202 Curth & van der Schaar, _ 202 l). gaussian process (Alaa & van der Schaar, (011)) and tree-based ap- proaches (Wager & Athey, 2018). A critical perspective in recent literature is minimax optimal- ity. Kennedy et al. (2024) proposes a minimax optimal CATE estimator based on the R-learner (Nie & Wager, 2020). by deriving a minimax lower bound (Tsybakov, 2008). While several di- rections exist for extending our results to CATE estimation, deriving minimax optimal CATE estimator would require further theoretical analysis, which beyond the scope of this study.
This study employs arguments for statistical analysis under stratified sampling (Wooldridge, 2001). which includes PU learning as special case (Imbens & Lancaster, (1996). Another example of stratified sampling is covariate shift; [eehara et al. (2020) study causal inference when the training data contain (covariates, treatment, outcomes) but the evaluation data contain only covariates.
The Heckman model (Heckman 1974)) and the framework of Robins et all 1994) are important contributions to causal inference with missing data. The Heckman model addresses endogeneity between the outcome and the observation indicator, whereas our analysis does not consider such
25

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endogeneity. Our setting is therefore simpler in that respect, yet more challenging because the relationship between treatment assignment and observation follows the PU learning mechanism. Similarly, Kennedy (2020) allow endogeneity between treatment observation and the outcome, but the observation mechanism itself simpler than our work. These approaches are not nested within one another; rather, they are complementary, and combining them may yield more flexible statistical modeling.
A.1 ATE estimation with missing data
ATE estimation under missing values has been extensively studied, the standard ATE estimation setting is itself closely related to the literature on missing data (Rubin, 1976; Bang & Robins, 2005).
In this context, various assumptions can be made about how data is missing. For example, some stud- ies consider settings with missing covariates (Zhao & Ding. 2024). This study, however, focuses on the case in which covariates are fully observed and treatment assignment is missing. In the problem of missing treatments, Molinari (2010) presents several examples from survey analysis. |Ahn et all (2011) investigates the effect of physical activity on colorectal cancer using data in which treatment is missing for about 20% of the units. Zhhag et all (2013) estimates infant weight outcomes where the geaamenn-mmothe's body mass index (BMI) is missing for about half of the sample. Kennedy (2020) develops general framework for handling settings where both the observation indicator and the treatment indicator separately observable. In contrast, in our case, we can observe only the product of the observation and treatment indicators, implying less available information than in Kennedy (020). Kuzmanovic al. (2023) proposes method for conditional ATE estimation under this weaker setting.
Our problem is also related to ATE estimation from misclassified data (Lewbel, 2007). Early econo- metric studies focused on continuous regressors (Hausman. 2001). With regard to binary variables, Mahajan (2006) analyzes misclassification in regression models, while Lewbel (007)) develops methods for identifying and estimating ATES under potentially misclassified treatment indicators. Researchers have also explored partial identification approaches when the exact misclassification process is unknown, providing bounds on parameters rather than point estimates (Manski, [193] 2010). In applied settings, validation data have been used to refine causal effect estimates under potential misclassification (Black et all 2008), demonstrating that even modest errors in treatment indicators can significantly impact policy conclusions. Yamane all (2018) also addresses a related problem.
Finally, we refer to semi-supervised treatment effect estimation Chakrabortty & Dai 2024). which primarily considers a scenario where two datasets are available: one with complete data and the other with only treatment indicators D and covariates X but outcome data. Although the setting is not directly related, integrating insights from both areas could enhance the applicability.
A.2 Introduction of PU learning algorithms
Another related body of work comes from the literature on PU learning. PU learning is a classi- fication method primarily designed for binary classifiers (though it can be extended to multi-class settings) in the presence of missing data, Its origins trace back to case-control studies with con- laminated controls (Steinberg & Cardell, 1192 Lancaster & Imbens, 1196). which are refined in du Plessis et al. (2015) under the term case-control PU learning. In parallel, Elkan & Noto (2008) investigates PU learning in the context of the censoring setting. One of the main applications of PU learnin is learning from implicit feedback, which commonly arises in marketing and recommender
systems. In such settings, user aetions-suchad as product purchases- are observable, but non- con-actions do not necessarily imply disinterest in the products; therefore, we might suffer bias in a classifier for predicting the users interests if we train it using such data with regarding the action and non-action as positive and negative data. As discussed in the introduction, we consider a similar application. However, our goal is to estimate treatment effects, rather than to train a classifier.
26

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B Reformulation of the censoring and the case-control settings
This section provides a reformulation of the case-control and censoring setting to deepen our un- derstanding. Note that the formulation described in this section is mathematically equivalent to the ones in Sections and 2.4
B.1 Reformulation of the censoring setting
We can introduce the censoring setting with the following story:
For each i, sample (X;, Di, Y;) generated.
A coin tossed, and Ō 1 if lands heads, or O = if it lands tails.
If Ô, and D, (that is, O,D, 1), then observe the treatment indicator
D. = 0,=0==1.
Otherwise, the treatment indicator not observed, and we only observe (Xi, Yi).
Finally, we observe (X,, O,D; Yi).
By denoting D, by Oi, we can obtain the same formulation in Section 2.3
Remark (Kenneyy (2020)). For example, Kennedy (2020) considers the missing treatment informa-
lion, which essentially different from ours. Kennedy (2020) considers the following setup:
For each i, sample (X,, D., Yi) is generated, where X; denotes covariates, D3 the treat- ment indicator, and Y; the outcome.
coin is tossed, and Ō, = it lands heads, or Ō, = 0f if lands tails.
-ff 1, we observe the treatment indicator D; 1, O along with (X₃, Yi). Of 0, the treatment indicator is unobserved, and only observe (X,, Y2)-
Finally, we observe (X,, Oi,O,,D,
For each i, sample (X;, Y;) is generated, where X; denotes covariates, D; the treatment indicator, and Y; the outcome.
Thus, while Kennedy 2020) observes Ö,, O,D,D, Y.), observe only (X,, O.D., Y.). In our case, Og itself is also missing (note again that (X,, O,D,, Y,) is equivalent to (X,, O,, Yi), where O ŌzD)). That is, in our case, we can observe only a subset of treatment labels with D; = while the remaining labels missing and consist of mixture of D, and D; O. In contrast, Kennedy (2020) can observe both D, and D2 = when the label is observed. Our setting designed to be more suitable for applications in marketing and recommendation systems, where implicit feedback is commor
Note that in our study, we do not explicitly use Ō. bul instead denote Ö₁D; by another random variable O,; that is, =
In machine learning terminology. we believe that the setup in Kennedy (2020) is close semi- supervised learning, where both a fully labeled dataset (X,, L3) (i 1,2. n2 and an unlabeled dataset X, (j 1,2, ..) available (Note, however, that in Kennedy (2020). whether data point is labeled is itself a random event, whereas typical semi-supervised learning assumes a de- terministic labeling process). PU learning is generally considered a distinct setling from semi- supervised learning though they are related. For a discussion of the relationship between these settings, see Sakai et al. 2017).
RReaarding the missingness mechanism of Kenneyy (222)) considers more general setting than ours by allowing Õ to depend on the outcome Y.. In contrast, while we do not allow such dependence, we use less information about the treatment indicator than Kennedy (020), as explained above. Thus, we cannot say which setting more general.
27

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B.2 Reformulation of the case-control setting
We can introduce the case-control setting with the following story:
There are two groups: the treatment group and the unknown group:
Treatment group:
For each j, a sample (XT.s, (1)) is generated and observed by us.
Unknown group:
For each sample (X,, D, Yi) is generated.
We observe (X,, Y.).
C
Details of the examples of ATE estimators in censoring setting
This section provides the details of the examples shown in Section
C.1 Example ATE estimator in the censoring setting
In the censoring setting, as we explained in Section 3. we can obtain an ATE estimator by replacing the following two quantities with sample approximation and nuisance estimators:
1[0 1|Y To X)
(1) E3
1[0 OiY
[o(1 x)1 1|2 9o X)=o(1 | X)
E|Y E[y E
E
[9 X)mo(0 x)
Such an estimator be defined as follows:
100 1|x
100 0|Y;
(1 x,) 1|y,
IPW
(0
20 X2)
In. (0 X2) (1, X.)
where न and In are estimators of mo and 9o- We refer this estimator as the inverse probability weighting (IPW) estimator, which is also shown in Remark 4.4
For this estimator, we can show the following theorem.
Theorem C.1. Suppose that Assumptions 3.2 3.3
and 4.4 If [ X) To(d X) op())
and 1 (d) X) go(d X2 = (1) hold as n + 0o for d {1,0} then PA Prens-IPW Pg To holds as n + oc
Proof. We have
1 Σ
1(0, 1|8
10, 110,=00 008;
(1 x,) 1|1;
gcons-IPM IPW
71 i=1
(1 X₂)
[n [0] X.) X.)
(0 Xz) X.))
100 = 1|y;
110 = 0|Y;
90 x,) = 1]x,
To X,)
(0 (.) X,)
0o X.) X.)
110, = 1|2
1(0, = 0|Y
9o X,) = 1|xi
(o X.) (1 X.)) ))
+
To(1 X.)
(0 T.) X.)
1[0; = 1|y
1(0 = 0
@n x,)
+
+
(1, Xx))
[n X, (0, X2)
(0, 11) X,)
100 1|
110 = OOY,
(0 X,) = 1|K;)
+ +1)
[1 X.)
(o (0) 0 X.)
go(0 X1) X.)
1=1
Here, from the law of large numbers, we have
100, 1|
10 1
P. E
( Xx)
X)
28

---

1[0, - =0y;
90 x,) 1|y
Oo
X, To X2)
9o X;) X;)
1 = OjY
9o(1 X) 1|x
E
E
0o X (o X)]
9o
X ) x)
Thus, the proof is complete.
C.2 Example ATE estimator in the case-control setting
In the case-control setting, PU learning methods have been investigated by Imbens & Lancaster (1996) and du Plessis et al. (2015). In that works, we typically make the following assumption, which corresponds to the SCAR assumption in the censoring setting.
Assumption C.2. It holds that (r,((x) = Co(x
1), where Co(x D = d)
(o co(d)
This assumption is also attempted to be relaxed by existing work, such as Kato et all (2019) and Hsieh et al. (2019) introduce their approaches. As well the censoring setting, various relaxations exist depending on the application, and there are trade-offs between the strengths of assumptions and identification (Manski, (993).
We further make the following assumptions.
Assumption C.3 (Unconfoundedness in the case-control setting). The potential outcomes (1), are independent of treatment assignment given covariates:
(Y()), X.
Assumption C.4 (Common support in the case-control setting). There exists a constant e. indepen- dent of n such that for all a E X, eo(d x), (T,o(r), Co(x) > e hold.
Under these assumptions, the ATE To is estimable by replacing the following two quantities with sample approximation:
E[Y(1) = E rY)))
and
E Tea(1 X)Y
B() = R
(o X
co X)
where recall that (o()
Co(X) [x)+ en can be estimated using PU learning methods, and expectations
can be approximated by sample averages.
Semiparametric efficient ATE estimation under the case-control setting
In this section, we consider efficient ATE estimation under the case-control setting. Similar the censoring setting, we first derive the efficiency bound and then propose an efficient estimator, provid- ing theoretical guarantees for its consistency and asymptotic normality. Throughout the arguments, we assume = an and | = (1 a)m, where a E (0, 1).
D.1 Efficient influence function and efficiency bound
Using efficiency arguments under the stratified sampling scheme (Uehara et all 2020), we derive the following efficient influence function (see Appendixū]for the proof).
Assumption D.1 (Regularity conditions). The outcome Y and covariates X have finite variances. There exists a constant C > 0 independent of n such that pT,o((), (X,)(X) E [-C,C].
Lemma D.2. If A'sssmpttios C.2 D.I hold, then the efficient influence functions are given as The (T) (X)Y(()) (1);; cn. ro) and spee (U)(,, Yu; pT..0, Ju.o, eo, To), where
eO (T) (T) (1): PT, Co. ro) = goe (T) (T) (():; Pr,0, ro); ce (x) Yu; WT, MT.00 TO) -====== = See (X) (X, PT. To.
29

---

S (()::)()) co. =
co(1 {x)
(r(1) - (o),(()) (oxxx))oo(3)) ro(x),
eo(0) X)
S (X, (", MU,00 Yu mu,o(x)
mu.o(X)
co(1 x)) co(0) X)
co(0) x) xTTO(())
co(0 X)
and recall that ro =
(o(X)
(T,o(X)
Then, we obtain the result on the efficiency bound.
Theorem D.3 (Efficiency bound in the case-control setting). Assuupttinss the asymptotic variance of any regular estimator is lower bounded by
C.2
C44
D. then
Vee
E Tee (T) (X,0,, L.O,Y,;ni 1T,,0 co,
ro)² (0
E
Thee (U)
(X, K,O,Y TY 60)2
a
1 1-a
E a
11
co(0 X) Var(Y (X)o(())
co(1 x)
+ -
Var(Ye (X)
+ (To(X) - -T0)
co(0) X)2
where a = m/n.
D.2
Semiparametrie efficient estimator
Based on the efficient influence function, we define
orc-off
S (T) (ii Sn Fr.))
Σ sec (Xx, Au En,a.
1=1
Here, ni MU,, En and Fn,i estimators of μT,00, MU,,,, co, and ro. where m and I denote the dependence on each dataset. Unlike the censoring setting, we do not use the observation indicator O, it deterministic whether unit belongs the treatment group the control group. This distinction leads to differences in the theoretical analysis compared to the censoring setting.
D.3 Consistency
We make the following assumption.
Assumption D.4. As + it holds that |ie..; eo -1((1) and || - roll2 op(1).
Then, the following consistency result holds.
Theorem D.5 (Consistency the case-control setting). Assumpptio C.2 C D.1
and D. holds.
then pra-eff P, Ty as n oc.
Interestingly, achieve consistency, it is sufficient to obtain consistent Tn Compared to Assump- tion in4 the censoring setting, consistency of the expected outcome estimators fAT,.. and D; is not required. This is because, in this setting, the observation probability can be treated known and 0 for each dataset).
D.4 Asymptotic normality
Next, establish the asymptotic normality of the estimator. Similar the censoring setting, assume that the propensity score en is known and obtain estimators proo and μito via cross fitting..
Assumption D.6. The propensity score ey and the density ratio To are known and used in construct- ing poc-eff i.e., Tn,i en and Fn.; ro.
Assumption D.7. For each l L, the following conditions hold as cor ||TT,o(X) (()) MT. (X)1g2 (1) and || (X)((X))1, ())
-18)
We establish the asymptotic normality in the following theorem with the proof in Appendix result, we consider the scenario where the sample sizes m and approach infinity while maintaining fixed ratio m : aa).
K In this
30

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Theorem D.8 (Asymptotic normality in the case-control setting), Fix a E (0,1). For 0, consider the case-control setting with sample sizes m. such that m an and a) Suppose that Assumptions C.3-4 D. D. and D.7 hold; that is, Cn, eo and Fn,i and
PTr,,. = TT.m' PU,ns P0, are consistent estimators constructed via cross-fitting Then, we have
Vn (T (7---et -T) To) S N(O,, VCo) as n n
Thus, the proposed estimator is efficient with respect the efficiency bound derived in Theorem D.3
D.5 Comparison with the censoring setting
Unlike the censoring setting, we do not require a specific convergence rate for the nuisance estima- tors if the propensity score is known. This is because, in the case-control setting, the group mem- bership-wheer a unit belongs to the treatment group or the unknown group-is deterministically known. This scenario can be interpreted as case in which the observation probability is known, meaning that only the consistency of the expected outcome estimators is required (Kato et all 2020 202
In other words, we can intuitively consider that in the case-control case, the observation probability is given as one for the treatment group, while it is given as zero for the unknown group. Since we know the true probabilities, we can ignore the estimation error unlike the censoring setting. Note that this interpretation may be mathematically confusing since it gives us impression that the case- control setting is a special case of the censoring setting where the observation probability is given one zero. This understanding is not correct because in the case-control setting, the treated and unknown groups are different datasets; that is, the sampling scheme is completely different. This sampling scheme has extensively studied as stratified sampling scheme from 1990s to 2000s by existing studies such as Imbens & Lancaster (196)) and Wooldridge (2001).
PU learning algorithms E
We review representative PU learning methods. For all methods, the goal is not to obtain a condi- tional class probability (propensity score) but rather to obtain better classifier. However, under specific loss functions, including logistic loss, the obtained classifiers can be interpreted as estima- tors of the probability (Elkan & Noto, 2008 Kato et al_ 2019 Kato Teshima. 2021).
E.1 Censoring PU learning
In Elkan Noto (2008), it is assumed that only a fraction of the truly positive instances are labeled positive.
Let O denote the event "labeled as positive," and let D indicate true positivity. In our study, O is called an observation indicator, and D is called treatment indicator.
First, we make the following assumption, which plays central role the method of Elkan & Noto 2008):
= D -1, = D Vz E X,
(1)
where (0, 1] is a constant (Assumption 3.J). Intuitively, represents the labeling probability or censoring rate, which denotes the fraction of positive instances that are observed (uncensored) the labeled dataset. If relax this assumption, may not pointy identify the ATE without different assumptions. There are various approaches proposed to address the relaxation (Bekker Davis 2018).
The learning procedure proposed by Elkan & Noto 2008 consists three main steps (for details, see Elkan & Noto )008)):
Estimation of mo: First, the observation probability mo is estimated using standard regression meth- ods, such as logistic regression.
Estimation of ct Next, is estimated using an estimator n, of np- Under Assumption 31 can be estimated by taking the sample average of म. over positively labeled samples.
31

---

Correction of the observation probability: From Assumption 3.1 we have To X) ) cF
1 lx).. Using this relationship and the estimators of To and c, P(D = 1 X) is estimated as (1 x
Remark. Violation of the assumptions The impact of violating assumptions depends on how the data deviates from the assumed conditions. For example, if treatment labels are not missing at random, the original estimator of Elkan & Noto (2008) may no longer be valid. In such cases, alternative methods
those proposed by Bekker & Davis (2018) and Teisseyre et al. (2025) m0ay)mmay may be applicable, although they rely on different assumptions. In the case-control setting, du Plessis et al. 42014) provides a sensifivity analysis of the trained classifier when the class prior is misspecified.
Remark (Time complexity). The computational cost of estimating the conditional class probability via PU learning is comparable to that of standard logistic regression. For example, in the censoring setting, Elkan & Noto (2008) proposes a method based on logistic regression. In the case-control setting, du Plessis et al. (2015) presents an unbiased PU learning approach which, under the log- loss. has the same complexity as logistic regression. Specifically, for linear- in-parameter models, logistic regression has time complexity of order O(dd?² + n²) log(1/c)), where the feature dimension and e is the target optimization accuracy. If Newton's 's method is used with T iterations, the total time complexity becomes O ((dn² + n² n²),), which dominates the final averuging step.
E.2 Case-control PU learning
different perspective is provided by du Plessis et al. (0155) and subsequent studies, often referred to case-control PU learning. In this approach, the labeled positive data follow distribution (T)) whereas the unlabeled data are drawn from Co(x), a mixture of positive and negative in- stances.
Let be a classifier. In conventional supervised learning, the classification risk is defined as:
R(h) = e. +1) + col)) -1),
where co(1) is the prior probability of being positive, and (4, +1) and (h, -1) denote the risks over the positive and negative distributions, respectively. Specifically, (h, +1) represents the expected loss when predicting class -1 while the true label is +1 in the positive distribution, and (h, -1) represents the expected loss when predicting class +1 while the true label is 1 in the negative distribution.
Since negative examples are unavailable, du Plessis al. 20155 re-express R_ (h, -1) as:
(1 - (--c((1)RR(0 (h, 11
Ru(h, -1) (1,,
where Ru and R4 (h.-1 denote the risks over the unlabeled and positive distributions, respectively. The term (h, 11) corresponds to the expected loss when predicting class +1 while the true label is -1 in the unlabeled distribution, and R4 (h, -1) I) is the expected loss under the same prediction and true label in the positive distribution. Note that (h, +1) and R4 (h, -) are distinct: they consider different true labels while expectations are taken over the same positive distribution.
Substituting the above expression into the original risk gives the following classification risk:
R(h) eo (1) +1) (u, -1) en (h, 1)).
sample-basse based approximation of this formulation referred as an unbiased risk estimator.
For example, using the logistic loss, the unbiased risk estimator becomes:
R((h) = c0 Σ
log (1 + ([exp((A(Xx))
j=1
m
Σ log (1 + exp(I(Xx))) - co(1) m
log (1 exp(h(X,))).
Then, can train a classifier as h == arg minnch N((h), where H is given hypothesis set.
Note that we assume the class prior (1) is known. Several methods have been proposed to estimate it under additional assumptions Plessis & Sugivama, 2014).
32

---

E.3
Density-ratio estimation
Since the density ratio can be estimated in the case-control setting, we introduce related meth- ods. Density-ratio estimation has emerged as a powerful technique in machine learning and statis- tics, providing a principled approach for estimating the ratio of two probability density functions (Sugiyama et al. 2012). Let X; Za E X' be random variables. Specifically, if { {Xx J are drawn from (x) and (Zg }= are drawn from (0(2), the goal is to estimate
ro(x) Po
do
directly, without first estimating Po( and qu(2) separately.
Estimating (x) and (o(2) individually can be challenging and may introduce unnecessary model- ing complexities if only the ratio ro (z) is required. By directly estimating the density ratio, more stable and accurate estimates can often be obtained, avoiding potential compounding emoos from separately learned density models.
Various algorithms have been proposed for direct density-ratio estimation, including the Kull- back-Leibler Importance Estimation Procedure (KLIEP, Sugiyama et al.. (2008) and Least-Squares Importance Fitting (LSIF, Kanamori et al., 2009). These methods typically optimize a criterion that ensures the estimated ratio closely approximates the true ratio in specific divergence sense, such as the Kullback-Leibler divergence squared error, which can be generalized as a Bregman diver- gence minimization problem (Sugivama et all 2012).
From the Bregman divergence minimization perspective, PU learning methods can also be seen variants of density-ratio estimation, as demonstrated in Kato et al. (2019) and Kato & Teshima (021)).
F Remark on the nuisance parameter estimation in the censoring setting
In the censoring setting, by applying the method of Elkan & Noto 20080, we can obtain an estimator
of (P X) from an estimator of no x)) PP(O = estimate
X). However, our objective is to
go X) P(D = XX P(D = 1,0 = 0 X) P(O 0 D = 1)P(( =
P(O 0 X)
PP(O 0 X)
rather than P(D = I
Let (1 | X) be estimator of PP(D X). We can then obtain an estimator of go | X) as follows:
(1
nn (1 X) = म. X)
where is an estimate of =P = 0 D = 1). Notably, under Assumption c can be estimated by taking the mean of R, X) over the positively labeled sample (O, = 1).
G Pseudo-code for ATE estimation in the case-control setting
We explain how we construct the estimators of the nuisance parameters in the case-control setting,
We can estimate μT,0 and puoo using standard regression methods, including logistic regression and nonparametric regression. Specifically, for estimating PT,0, we typically use the dataset
{ (Xr,, ((()))) while for estimating μU,o, we use {Xx,, )
To estimate eo. we can apply case-control PU learning methods, such as convex PU learning pro- posed by du Plessis et al (2015). For estimating ro. density-ratio estimation methods can be em- ployed (Sugiyama et al. 2012).
Notably, (T,O (x) = (o d = 1), then To can be estimated from an estimator of eo using the relationship ro 1/(en(1 (1))
In cross-fitting, we split Dr and Dt, respectively, as performed in Uehara et 2020 The pseudo- code is shown in Algorithm G
33

---

Algorithm 2 Cross-fitting in the case-control setting
Input: Observations Dr {XT.,, Y5(1)) and Dr: {Xxx Yu,e) F k= number of folds
L, and estimation methods for fAr,o- PC,0: co, Tu-
Randomly partition Dr into I. roughly equal-sized folds, (F(C))ECC. Note that USEC J(t) = D. Randomly partition Du into L roughly equal-sized folds, (K(C))EE Note that ULec K(O) D.
for E C do
Set the training data as J(-)) {1,2. ... n \ J(
Set the training data as K(- {1,2,... n K(()
Construct estimators of nuisance parameters J( 4) and K(
Construct ATE estimate in (f)- using J(0, 1daand the nuisance estimates
ple) (P) PT, PU en .P end for
Output: Combine (Fnt-ef((() Jeec form F.c-eff
H Proof of Lemma 4.2
We prove Lemma 4.2 Our proof procedure is inspired by the one in Hahn 1998).
Proof. Recall that the density function for (X,0,Y) is given as
pu(z, o, y) = co(2)) [To(1 x)Pr(1).00)) m)) 1/0= 1)
(nu(O) x x)py .o (y mr) (/o=0)
where Py x). (o x), and py (y x) are the conditional densities of (()), (()).
and Y the censoring seting.
For this density function, consider the parametric submodels:
paub (Po P: PE
where Po has the following density:
ou( (o 0) To( (0(c:))(T(o z; ()pr()))) [ 1|0=11 (To x: 0);0pp(v x: xx [[o=0]
while there exists Og IR such that
p(z,0,y:0n) po
Then, we define scores as follows:
i log log
S(x,0, 0) ae
#(1 2; =Sx(x;0) 1l0 1] (y x;;0+
110 0] (Sp(s xx
# 0)
7(1 x;0)
= x;; 0)
where
d
Sx(x;
log ((2; 0),
d0
d
Sy (x;0)
log ( x)
dg
d
Sy(y
(x:0)
log py (y 1))))
do
d
#(0
x; 0)
(o 0).
do
Let T (S(x, 0, y; 0)} be the tangent space.
34

---

Here, note that
P x; = 90 x)pr(1) 2;0) + go(0 x po) x;
We have
paly 12;0) 9o x)
py (y x5 - go 3)pr x;())
Using this relationship, we write the ATE under the parametric submodels as
T(0) = /ʃ y (())pr x; (((;
//
()d
// (=
// 5(0) (o x))
py (v(0) x; (x; (1)0)))0000))))) (O)dd
/f ((0)
g0 x))
(v x; (2;; O)dg())))
90
Them, the derivative given as
θT(0)
Eg (I)) [x;0] -Eg
1
ySy(Y xx:0)
00
go x)
[9([1 X)
+ Es
(x))
900( X)
+ Eg
((X:0)5x(X:0)
where
T(;00= = #(1 x) go(0 x) U( IX)+ 90 X)
9o x) (T(() = =0(11 ((11x) mc(X)
From the Riesz representation theorem, there exists a function such that
θr( 00 8-
E [X
There exists a unique function Trens such that Nrcens T, called the efficient influence function. We specify the efficient influence function as
yrenne (X, S,OYY Y; PT,00, Po, To, 9o)
Scens (x, O, Y; LTT 0, No: To) To.
110 11(x ( - -
110 = 0/ 00(r -vo(x)
To X
9o ())To(0 X)
go | x) = 11(r--rr,00 )r,,(())
90(0 X ()oo(1 XX)
+ PT, 9o x) v(X) + go X)
To (0 X) (T) To-
We prove that Treens (X, O,Y; uT,o, Mn, ma, 90) is actually the unique efficient influence function by verifying that Treens satisfies and Treus E T.
Proof of (2): First, we confirm that freess satisfies 2 We have
Chue (X, O, Y: KPPro, Mo. To. S(), 0,;
T cens (X, X.O.X. Y: Pr.0, My To. To)
35

---

(Sx(x:0)+110 110 (5)()(() X:
| + #(1 [X;0)
M(1 x;
+ +110 0f (Sp(Y) xx + #(0 X;
x(0) x;00)
110 [110=110 11 - )))(()))
1 0/ - vo(x))
R
To(1 X)
90(0 X (o x)
go x) = -
90 X X) x)
1
((X) + 90(1 XX)
+ +)TO(X)
90 x) PT, (X) - -T
9o x)
(y(i)( (x + #(1 (x;0)
(xxxx))+000000 1]
x(1 x;
+ 110 = 01 (5>(Y) xx;) + +(0 Xx;0)
x(0 X;
E pT. - 9o x) ((x) + 90 X)
@o xx UT, (XX) - -T Sx
110==110 = 11(r f0y--PTAN pTo(X))
110
go(1 X)1 0 11(r 1 -(T,(X))
To(1 (X)
0o 11 1x)
+ (T,((X)
9o xx (XX + 90 X)
1
9o x) PT,
110 11 (Sriy( (x;00) +
#(1 (X
T( Xx;
110 =10=00( 11y--u -vo(x))
9u(0 x) (XX) 90 X)
+ (r,o(()
(0 x) (T,,(X) 50
90(0 ()TT((0) (X)
10 =0| (5x((r) (x:0)
#(0 x;00)
(00 xx
where we used 110 = 1]110 = 0| 0, and
1 100 1] 110=11(--770(xx) -
110 1] (Y(1) -(ii)))rr,((()) (x))
To X)
To(1 1 X)
[1 x) ())(((r,)) (rr (xT)(X)
=
=0,
To(1 1x)
1O = oj(Y - x
= [ = (8 -V
x x) (v6(x)) (xx( vo(x))
E
E.
= 0,
00 (0 X)
90 X)TO(O X)
90 | X)TTO(0 X))
90 X)1 = 11 -xT,(())]
[o | X (1 X)
(UT,o(X) - (x))
E
9o o)=o X)
go | X X))o(1 X)
We have
1 px (o x) v(X) + 9o(1 X))
90 x) (T)) - To (x(x))
36

---

(110-1(x-
(100 = 1a0011X-pruaa pT,(x))
9o(1 [o(1]x)10==1]] -1((r - -)T,(X))
To x)
9o X))o(1 X)
+ (r,o(()
90 x) v(X)+ + 90 x)
90 xx) (r,((X) - To 110 = 1| (Srco(r( 11:00)1)0)))))
(x;6) + #(1 | Xx
tT1 X;
110=0 - -v(())
1
((X) + 90 X)
+ (TT,( 9o x)
90 x) 1 (TT,o(X) - T0
90(0] (0) (X)
10 -0 (55(Y) (x;)
#(0 X; X; 00)
<T0 X; 00)
90 (1 X)
E
[TT,((X) 90(0 x) (X)
( X) /xT,0(X)) Sx (X;
110 - =11(1-1 10y-ur,oo pr),())))
9o x) (x-
Sya)(Y (x)
T0(1 X)
(0 | x) (X
100 -oo()(((X))))
Sx(Y (x
90 (0) (X)
[TT.0(X) 90(0 x) v(X) + 01 x))
E
90 x) (T,,(()) Sx
100 [ (1 90(1 X) 11)
100 1 0j 90 x)=o(0) x) 5 x:00)
(x:)
To x)
90(0 x) x)
where we used
(x; =0
(UTO(X))
9o x) v(X)
+ 90 X)
90 x) -
110 11 11( (Syas( | (x:0) #(1 | xx(0)
m(1 X;0o)
0.
Finally, we have
(TT,O(X) 30(0 x) (XX) + 90 X)
1
90 x) (x)) Sx(X;
+
-110= 1|)()
To(1 X) +
90 x) = 1)
(F()) | (x;6)
9o () XX) Sx xx;
1 = -
9u(0 X {) x)
E ((1))r()) X Bb)] [90(0] x) (T | xx00
[9o[1 X) Y (1)S)(()((())) x:
+ +E
[90 X))
+ Es,
(xx;0)Sxx (X:
00
0=0p
Proof of grenss E TT: Set
Sy 2) y- E (Y(1) X =
To x)
37

---

Sy y E X = x
To(0 xx)
90 X) 90 x) () TO-
0x(X;0)= (x),(x)))) 9o(O x) v(X)+
Then, yreuus E 7 holds.
Remark. Some regularity conditions are needed to guarantee that the expectations of the score In particular, must justify interchanging integration and differentiation for the relevant densities. This holds, for example, when the density is integrable and has a bounded first derivative.
functions
standard to construct a regular parametric submodel of the true distribution. Regular parametric submodels are technical tools used to derive the efficiency bound and can be constructed to satisfy the necessary regularity conditions. For true density Po
More precisely,
typical con- +t((t))))) (()))))o(()) where g(z) is bounded score function and t E [0, xo). This BE t=0 log (c)). If g(x) is unbounded, we can still define parametric submodel where l is nonnegative function satisfying k(0) = k = 1, and e(t) a normalizing constant ensuring that Jf pe(r)dd 1. For more details, see Section 25.16 of van der Vaard (1998).
struction is ps (x)
satisfies g(x)
as (x)) (t)x)((g())))llll
In our study, we verify the regularity requirement by checking that the parametric submodel induced by the derived score satisfies the necessary properties. In our case, the required regularity holds under finite first and second moments of the outcomes (Assumptions 4.7), together with the common support condition (Assumption 3.3)
I Proof of Theorem 4.8: Semiparametric efficient ATE estimator under the censoring setting
For simplicity, consider two-fold cross-fitting; that is, L = 2. Without loss of generality, assume that the sample size n is even, and let = n1/2. For each l {1,2}, we denote the subset of the dataset in cross-fitting
DIO) = 1(x; :,00 ,000,,000000 )
We defined the estimator
1
ccens-eff
Scens
(X,, O₁, Pr Priis Rnss 90).
am1
where recall that
goens (x, O, Un An.is 9o)
10 = 1s(r (x)
10 -Y- go x)n = 10010=1[[r--0 Tur -I
+
Rn,i(1 x)
(0 x) X)
(0 X X)TT,, X)
90 |X)
+ (X))
90(0 X)) v(x)+
00 X) (X.)..
We have
pcons-off I
1
Σ Scons (xona(X,, O,, Yú; PT.m Dn,s, Rn,i 9o)
n
1 Σ
ocens (X,, O,, Yi Po. no. go)
Stens (X,, O,, MT, Vo. no, go)
gcens
1=1
Scens [x,, O,, Yis AT Pa,is go).
38

---

Here, if it holds that
n
S ceaus (X,, Oi, To.
Seeus (Xi, O₁ YG; AT Pn, an,i. - Op(1/vn)
1=1
(3)
then we have
1 n To)
poens-eff
Sceus (X;, Oi, V1; PT,,0 Vo. To. + 0p(1)
n
vn
4 NN Voees)
from the central limit theorem for i.dd. random variables.
Therefore, prove Theorem 4.8 by showing 3 We decompose the LHS of (3) as
Sreus (X;, Oū, Yi; NT,0, "0, To. TO.000 9o)
geeus (X;, Oi, Ys; NT, Dn 90)
π0
Σ
Scens (X! 0.00 y pT Mo, go)
T1
71
(1.2)
1
Scens (0) 0 y
(2) MT 110) 1,11,,000 .9.2..)))
Let ( denote the é-th fold of D. Here, we have
S ceus
( UT.O No. no. 90)
Seens (x (0) 5 (
PT
(6 ((0) =00 go)
m
m
Scens
X
PT,0o ub, no: gtens (x(() -000 7:00,
PT
19 20
gcens
v (T,,0, vb m0. DD0]]
E gcens [ 0.0 7:10 ( 2(0) 7/ .90) [D0]]]
+ goens 1x) 0000 10 pr.o. Mo. Tu. 9o) D(0]]
E goens (x 0.00 y PT. 200 #(0 .0)) D(O))
To show
3
we show the following two inequalities separately:
m cens
m
5:(0 pT,o, Vo, No. 90)
Scens (x 50 ,. -0) : )(0,((0)
n
Scens x
v100, MT,0, 40, no. 9o) DD0]
E E geens (x 5.0) ((0) ( 110) 7( .00) D(O))
= (1/V/m))
(4)
E Soons
(x,0) 5(0) V pT.,0, No. no. 90) D())
39

---

E gcens (x,") 0000 " . 1 .o) DD()
-,,(1/mm)
(S)
Here, the LHS of the first inequality referred to as the empirical process term, while the LHS of the second inequality is referred to as the second-order remainder term.
L1 Proof of 4
Proof. We aim to show that for any E > O,
Scenss ( 0,00 y:(0, UT,,0, N0
lim P
#2000
m
Scens (t) -(00 -:00) TT.: ((0) #/0 .9))
72
-E E
{ gcens x(0) 0 y10) μT vo: To 90) D(0)
E. goens x (1) -.t0 7: Tf Di0 # .0) ((0) 2e)
0.
(6)
We show (6 by showing that for any >> O,
seess (x:0 y:10, PT.O. 40 To.
lim P
0 y,(0, PT
(000 1/ .90)
E [ goons (e) ( (
pT.. Vo. To. 90) D(!)
E geens ( 000 y T 1/00 - .90 D(O) ) 2 D(O)
0.
If holds, then (6) also holds from dominated convergence theorem.
We prove using Chebychev's inequality. From Chebychev's 's inequality we have
m
geens
(X O(.)) (:(0) pr, Mo, To, go)
s senss
(X
5.
πt 1=1
n
R
geeus
(1() 7(()) 7: pr.o, vo, no. 90) DDO]]
E
gcens X
v:10 PT.. 200 210 .00) D(O]
E D(1)
77
Var
- Swens (X (0) ,.00 (1(0)
PT. t0, To 9o)
1 m
goens (x(1) ,0!0 ,,10, :200 200 #00 (,0)..))
(x
7 1=1
E
conns
(x((0) (000 y,(0), pr, Vo. no. gu) DK0]
40

---

18 gcens {x , 1 200 -
)
Since observations are i.i.d. and the conditional mean of the target part is zero, we have
Scens (1) , 1 PT. Vo, Tu, g0)
Scens (0) 10000 , p _
mVar
77
n
E
(1) 000 (0 pT,,,, Mo, no. 90) D(O)
- E gcens X ,. 11 PT. , .0) D(o)] (01)
(
Var goonE (x!0) _ (1(1), UT,,0 vo, To, 90) - Scons Scon (1) 0:0 , Tn 5/0 1,, go)
B geens (x(() 0 v. MT,00 "0, no. 90) D(O]
:: {00 PT. 110, 100000000000 .o0 ((oaa locc
E gcens
geeus (eens(x(") ,.t² v.(); UT,O, vo: 0o. 00. Scens (x ,.00 8.0. ,,, 2(0 .0)
R
(8)
[ geeus (x!0, 0.00 y/ JT.,0: vo: no. 90) [D(0]
(0) T. 239, (0, .0) b(0) )011)) 2
- E grons (e) ,. v
D
The term H converges to zero in probability as + xo if || - - = (1), 1, =
(1), and Wo - # ||2 op (1) as + co. Here, we used the boundedness conditions of each function and the following computation. Them, we complete the proof.
We explain the last step of the above proof below. Let A and B denote the first and second terms in the expectation of ( respectively. Then, we have
E B - E A
[(A
[A- B (I-BI(())
|D10]
Here, we have
(A B)2 DDO] - (E A - B 1(0))² S S B)2 DIo].
By showing that E [(A B)2 DDO] =1,(), we prove the statement. [(A- - B)2 D = o,((), we use the following concrete form of gvens,
To show
Scens (X, O, Y; U, .,)
10 = 11(8- (X)) 7(1 X)
10 = oj(Y - -v()))
+ 9(1 X)110 = 1J(Y - (X))))
g(0 X)T(0) (X)
g(0 x)T(11X)
1
v(X) + g(1 X)
+ (X((
9 X) MT (X).
g(0 x)
Then, have
A B
41

---

10 1Y - (TT((X))
110 = 0J(Y mo(x))
go | x) = 1J(Y pT)(()))
To(1 x)
90(0 X) {))o(( X)
90 xToo x)
20 X)
+ + 90(0 x)
Vo (X)
90 X) M (T,)
110 - 1 "T. (X))
1
Oi(Y - 1// (X)) + 90(1 X)1(0 - 11(Y - -T
2 (1 X)
9o X x))n((() (0) ()
go(0 X)m
(8) (1 X)
(0) + PT (X)
1
1 (X)
90 X) (1)
X) P PT. (X)
go
X)
90
Here, we can show that the following term converges to zero in probability, which follows directly from the convergence in probability of each muisance-paramete estimator:
(TTO(X)) - 5o(0 x) vo( + _o X)
0o x) (TTI(()))
( 9o x) (00 (X) (X) + 00 X)
9o x) H (xx))
Then, we show that the remaining parts converge zero in probability. Let denote the parts
(+) = 1 1[ (T),(())
100 0| vo()))
90 X) = 1](Y (T.,(()))
To x))
(o mo Xx)
go X)mo(1 [ X)
110 11(Y - - 110 = 0|(Y - 2/ (X))
go X)1 1](Y - -T (x))
a(6)
(X))
(1 XX)
00 X x)a (0)
(0) X)
9o X) (1 X)
Next, we have
(x) 100 = 1| - (T)O(X))
100 = 0| ( - vx())
9o(1 X)1 = 1](Y - uT.O(x))
To(1 X)
90 x0 X)
9o X) (X)
100 1](8 - -MT,o())
10 (j(Y - (x())
+ 90 X) 1] 1108-1111 (x,,0()
2 X)
go | x))((( (0|x)
90(0 x) x)
10 1[ (X))
10 OY (x()))
+ 9o(1 X) 1] (X))
2 X)
9o | x))! (0 ()
90 Xx)x X)
10 1[ - -p (X))
10 = 0 - 1/00 (X))
go X) 1] TT (X))
T |
9o x)* (0) (0) (X)
9o X)=! (1 X)
Then, from the parallelogram law, we have
100 1)(Y - (x)) 2
1 1](Y (X))
(x)2 2
T0 1 x)
# (1 X)
110 = o| vx()))
1 = 0| - vo()))
+
(0 X) n (0 X)
(0 x x) (0) X)
+...
go X)1 = 1](Y - (x)(())) g0 X)1 1](Y
PT (X)) (x))
2
+
(o x)#x() (1 (x)
9o X X) (1 X)
Here, we can bound
( X)1 = 1] (T,,o(X) 90 x)1 = 1](Y 1| 20 (X)) 2
2E
(D())
(o x) X)
go X) x) XX
42

---

by
(2) CE((TT((X)- - T (X))².
where C > 0 is constant independent of and we used the boundedness of 9 and म. Similarly, can bound each of the remaining terms. Thus, we complete the proof.
1.2
Proof of 5
Proof. We have
E
ceeus (1) ,0000 y: pT,,,, Mo, No.9o) 90) [(6)] [ (0) 00:0 ((() (())
1 : 1/ _x(6) .00 (DO0]
[1 11( Y -),,o(())
110 -10=0| 00 - -oo())) go(1 X]1(0 - (
E
+
To(1 | X)
90 x)=(()) x)
90 ( X)
g0 X)
+ (X)
90(0 x) (X(X)
0o X) UT, ( (D(6)
110 11(r y -
(6) (x)
110 0| Y 2/60 (x)
E
# (1 X)
9o X T (0 x)
90(1 x)1 = 1| Y PT (X)
(0 X x) (0) (0|X) X)
+ 7T (X)
1
(10) (X) + 9o(1 X) ((0
90 X)) TT,,,( (x) D(E))
(0 X)
+ (0 X)
E
(x)
90 X) (x(
90 x) (X))(())
TTo[1 X)) (KT,o(X) -1
#o x) (x() - D/ (x)
IE
#10(11 [1| |X)
9o X x)=xh((0 (0) X)
go(1] X))o(1 X) [TT - (x)
(0 X) x)
(1) (0) x)
p4) (X)
(0 X) ((0()) (X) + 9n(1 X) (1)
1
9o x) PT (X)
To(1 XX)
(ML)(()) 210 (x))
E
200 (1 (1l x))
To X) (21 (X) (x) (M))(x))m(()) mo(x))
+ E
9o X) (x)
1
(X(X)
90
X)
(0 X)Tl('(0 (0) X)
[o | X) To Xx) [UT,O(X) pl2 (x)
90(1 X)
90(1
X)
+ IF
0o x) (x,
"I,n (X)
((6)
90
x)
0o X) x) (0 X)
To(1 x)
(NT,,(X)) M4P (x))
E
-00) (1 (x))
1
(0 (0 x)
(x) (((x))-m(() (xx)))))(x))
+ E
1
9o X)
(1) (0 x)
43

---

90 x)
(1 x)
(NT,O())
ii -
1
90 X)
7/ (0) xx)
(1 x) - mo 1x))²
(MLLO(X) - (x)((()))
C
E
E
(1()) (0) x) - mo x
(xxx))))))²
+ CV
E
+ C
E
( (0) x) mo x)) E
- (x)) |
- -
= (1/vm/
where we used Hölder's inequality.
1.3 Remark on the double robustness
Indeed, a key structural feature specific to the PUATE setting is that double robustness holds only when the propensity score is correctly specified. That is, in the PUATE framework, double robust- ness can be stated as follows: given a consistent estimator of the propensity score Go(D X), if either the observation probability (e.g., 7(O X)) or the conditional expected outcome (e.g.. UT, oX) (1) X] and (c)o(X) = (0) XD is consistently estimated, then the ATE estimator consistent.
This arises because estimating the conditional expected outcome for the control group. (X) = E|Y (0) X]. requires the propensity score go (D X). In the standard ATE setting, E[Y X]] can be estimated directly via regression on outcomes with D = O. However, in the PUATE setting, must rely the following identity recover (Y) X].
In the standard doubly robust estimator, (0)] can be estimated by taking the sample mean of
D. 0J,Y, - (xc(x;)) *,c(X,))
7(0 X,)
In contrast, for PUATE under the censoring setting, E[Y(O)) can be estimated by taking the sample mean of
10, = 0]6, ((X))) T(1 X,) 1](Y - (T,,(X.)) (X.)) 0(0 x,) X,)
7(1 X,) 3(0 X,)
((X))
(X,)))
2(0| x,) X2)
2 X,)
Put differently, the standard ATE setting, (O) can be estimated either by the sample mean of
1|D OjY
7(0 X)
or by
Te(X).
One relies solely on the propensity score g(0 X). and the other solely on the outcome model
(x(X). However, in the PUATE setting, E(((O) can be estimated either by the sample mean of
[[0, OjY;
0( X,) = 1]
@ X2) X2)
g(o x,) X,)
or
Ø(1 X,) pT (X.).
((X))
0 X3)
7 X,)
Both estimators depend the estimated propensity score ÿ.
This highlights the intuition behind the lack of double robustness in the PUATE setting: to esti- mate (or identify) (0) X]. one must necessarily rely estimator of go- This structural
44

---

dependency underlies why double robustness, in the usual sense, does not hold in PUATE. Even in cases where the treatment indicator contains missing values, if E X] can be estimated without relying on G. it is still possible to construct doubly robust estimator involving the product of the propensity score error and the outcome model error. However, the PUATE setting, we observe only treated units and a mixture of treated and control units. This makes the estimation of B| (0) x] particularly challenging, Also see Dahabreh et al. (2019).
Proof of Lemma D.2
Our proof is inspired by those in Uehara et al. (2020) Kato et al. (2024). Uehara et al. (2020) re- visits the efficiency bound under the stratified sampling scheme, a generalization of the case-control setting, studied by Wooldridge (2001) and Imbens & Wooldridge (2009). In the stratified sampling,
we define an efficiency bound by regarding 'rr, Pr) as one sample.
Their proof considers nonparametric model for the distribution of potential outcomes and defines regular subparametric models. Then, (i) we characterize the tangent set for all regular parametric submodels, (īi) verify that the parameter of interest is pathwise differentiable, (iii) verify that guessed semiparametric efficient influence function lies in the tangent set, and (iv) calculate the expected square of the influence function.
In the case-control setting, the observations are generated as follows:
Dr ((())) (Xr,, (roo(,,y(()) (r,0(())) w),
Du = (u.) Yu,) ~ pu.o(a,.))) = (o(x))rr. x)).
We derive the efficiency bound by regarding
E = (Dr,
as one observation.
We define regular parametric submodels
ppub = (Pr,0. Pu.o: 0 RR,
where Pr.e is parametric submodel for the distribution of Xr.. (1) and PUço is parametric submodel for the distribution of Xx, YU.k-
We denote the probability densities under Pr.o and Po.o by PT(x, yi 0) = (T(2; 0 P) (1,,((1))) x);), (u y: 0) = ((x; 0) (vu 0)
We consider the joint log- likelihood of Dr and Ds, which is defined as
m
log (PT(XT,) (((1))) Σ log (Pu(Xe, Yu.u;)).
By taking the derivative of y= log (pT (Xr,j, ()) + Ek log (pu (Xa, Vu,x:0) with re-
spect to 8, we can obtain the corresponding score as
d
5(E;0) de
Σ log (XT,;
Σ k=
log (Xu(Xk, Yu,u;6))
m Σ Sxr (Txziii0 + Σ
Sya)( XT,ji + Σ Sx(x;; + Σ Syu I (xx:0).
f=1
where
d de
(2;
log (x
45

---

d
Sya) 2;
log Py (1) (vl x;
d0
d
Sx(x;0) = db
log ((x; 0),
d
Sy x;;) = d0
log Pyu (v x;
Let us also define
See ( (x; Sy x;0), ge0 (1) Hc) Sx (x;
Sy (y x;
Here, note that
PY (y 2; co(1 x))r(1)( xx; co(0 x) 2;
We have
(y 2; eo x)) pru (y x; co(1 2)pr()(()) 2; 22:))
Using this relationship, we write the ATE under the parametric submodels as
r(0) = // u (())))()(((()) x; O)d((()
l/ y (0) (v(0) x: ()dy((()))
// (00
// y(0) co x)
PYu (v(0) x; (x; ())y
+ Iʃ
co x))
v(0)
Py (v) -; 0)
eo
x
The tangent space for this parametric submodel 0 0o is given
T
Sx, (XT,ji 00)
Σ (x Σ Sx(Xx: + Σ SYu (Yo,, (xx:do) E La(e)]
From the Riesz representation theorem, there exists a function such that
θr( E (E))
(9)
00
8=00
Here, we use the assumption that each random variable has finite variance, which guarantees the existence of the function.
There exists unique function qhe such that The T, called the efficient influence function. We specify the efficient influence function as
T (E; pT,u, MU,u, Co. ro),
(o X1.s) (Y5(1)-Prodl -(r((x)) ro(XTss), co(0 XT.i)
Yuuk - -M,,((x))
eo(1 Xx) en(0 Xx) 1T, (x.)(((s))
(Tx)(Xx)) co(0) Xx) uu.o (X)
co(0 Xx)
We prove that T
(X, S. Pr,,0 Pb, no. go) is actually the unique efficient influence function by verifying that oi satisfies 9 and th e T.
46

---

Proof of 9): First, we confirm that The satisfies 9 We have
E The Pr,D, yo, no. 90;; 0u)
E
(E; 10. No, To, 90)
Sx, (XT,ji + Σ ( + Σ Sx(Xx; + Σ Syu (Yu (X2a(8h) )
k=1
(1
eo(0) (1-000
co(1 Xx,})
(x)(1) (o(()--ar(A8) pr(X(s)) ro(XTs)
E
XT.,)
==1
Yuux - (u,o((x))
Mu + ea (xx)
+ (T,,(()))) Eo(0) Xxa)
uT, (Xe))()))))) -70 -
Eo(0 Xx))
eo(0) Xxx)
k=1
m
Σ
Sx, (XT,i 00) + j=1
Σ ==1
(ya)()()))
Σ Sx + Σ Syu (Yu, |xxi (xxilh)) 0n)
Since DT and Di are independent and observations are i.i.d., we have
co(1 XT.,) eo(0) (T..))
(x,(0) - pr(Xes))) ro(X1.))
1
m
j=l
Yu, (u,o(X)))
+ #T,o((X)) co(0) Xx) uu (Xx) + ea(1 [xx)
UT (XT,)(Xs)) - TO
7
co(0| xx)
eo(0 Xx)
k=1
m
Sx, (XT,; 00) + Σ (y())(((11) XT); (r))))) Σ Sx(xx;) + Σ Syu (Yu,. (xx:Bo) )
m
k=1
c1 Xr,3) ) (TT())))))((x))00( (8x, (XT,; 0o) + (r(1)())())) (xTx;
E
co(0 XTu)
(Yu, - (x,
+ (r,o((x)) - eo(0 xx) (u,o(Xs) + co Xx)
+ E
(0r,((Xs) - -oO
co Xx))
co(0 Xx)
(Sx: + Sy, Xx:
Because the density ratio allows us change the measure, we have
1 c1 XT.))
co XT.y) )
(T.O(XT))) ro((T.) (Xx,(i)): Sr()((((11) | x;(6))
(o x) (r(1) - -T.(())) (Sx(XT):)b) (Y())(Y x:
eo X)
E
Finally, we have
(1- co(1 1x))
E
co(0 (
+ Sy (xxxx)))xxxx;0)) ( (X) X:00))
(Yu (Yu- - -U,o)))))))
+ (T,((X) eo(0 x) Mu( + ( | X)
+
co(0 x) uT, (TT,(() - To
47

---

(Xx Sy (Yu xx))))
1
= Bo (()) xx -18g
Yusr (Yú xx)
ee(0 x)
[e XX)
Eg
(X) xx)0)
ca(0 X)
1g
((X;00)5x((;001)
(0)
00 16=00*
where we defined ( cotorx) co(21X) the second equality, we used the followings:
(r,(X) =(X;; 86). and from the first to
( 1 co x))
((())) 1x X
co
xx)
= 1 en X)
0, - pT(X) X] =0,
(0 x)
Yu mv,o(X)
E
Sx (X;00) =0
co(O X)
E
To (Sx(X; 0o) + (Yu
X;(0)]] = O.
Proof of The TT: Set
50c (T) (X) co. ((1
co x))
(V() (V())~TTo(()) mxo(x)) r(x).
co x))
YYu - mv,o(X))
+ +rr,o(x) co(0 x) (u,o(X)+ + co(1 X)
sec Yu: PT:
eo(0 x) (r,,(X),
co(0 X)
Then, The E T holds.
K Proof of Theorem D.G:
Semiparametrie efficient ATE estimator under
the case-control setting
Recall that we have defined the ATE estimators
pre-off
Sec (T) (X Yy; MT (10 Σ
200
s (t) (X,, Ye: Pl. P o
/=1
k=
We aim to show
Vn pffc-eff -T) 4 SN(00 Vog as 00.
Recall that
See (T) (X, ((); (00) 2(0), p(())
=
- -(1- 200 X)
X
(Y(L) - p40 (x) (X),
See (X) Yu: PT. (P)
i e(0)
Yu m (X)
-00)
On (1 X)
+ P (X)
Uun (X)
(0)
(0) (X).
En (0) IX)
(
240 (0) X)
(
(0) X)
48

---

We have
m
poc-eff
5 (T)
(X,; Yj: 10
200) ,00
Sec (X) Vii AU 200)
T,
m ==1
k=1
m
50c (T) (X,; HT,00, (0,
Sec (U) (xx, /v,,0
m 1=1
50c (T) (X;, NT,0
5e0 (U) (Xx, Yasjuu,,,
k=1
scc (T) (X,, Yji
Pj (
T. 2 (())
5e0 (U) (Xx, 1m
1=1
Here, if it holds that
s (T) (X,, Yj: (1)
MT 210 ((0)
1
S (T)
(X,. eo, = op(1/vm), (10)
==1
see (U) (xx, Ys: (() )
Σ
See (U) ==
(11)
then we have
poc-eff
n
T0
m
vn-
ge0 (T) (X), Yjs HT,,0 +o)+ + vn
Σ gee (U) (Xx, Hu,o, + 0p()
m
j=1
m
1
gce (T) Yyi co.0.
Sec (Xp, + (L()
am
((1 or)
j=1
4, (O, V(),
from the central limit theorem for i.i.d. random variables.
Therefore, we prove Theorem D.& by establishing (10) and I.. These inequalities can be proved in the same manner as the proof of Theorem 4.8 and the analysis of double machine learning under
the stratified scheme presented in Uehara et (2020) Since the procedure is nearly identical, omit further details.
L Additional results of the simulation studies
This section investigates the case in which the expected outcomes and propensity scores follow non-linear models.
All experiments were conducted on a Mac computer equipped with an Apple M2 processor and 24 GB of RAM.
L.1 Censoring setting
We generate synthetic data under the censoring setting, where the covariates X are drawn from multivariate normal distribution as X Co(x:), where (o(x) the density of \ I,), p denotes the number of covariates, and I, is the (p x p) identity matrix. We set p = 10. The propensity score is given by (1 X) sigmoid(X X [, + X (22, where X is the element-wise square of X, and 31 and (3 are coefficient vectors sampled from (( 0.572p). The treatment indicator D is sampled according to the propensity score.
49

---

Table 2: Experimental results. control settings, respectively.
The upper and lower tables are results in the censoring and case-
IPW
DM
Efficient
IPW
DM
Efficient
Censoring
(estimated go)
(true 9o)
MSE
6.86
0.51
0.28
2.30
0.17
0.21
Bias
-1.60
0.40
0.22
0.33
0.10
0.04
Cov. ratio
0.81
0.18
0.76
0.96
0.29
0.94
Case-
IPW
DM Efficient (estimated ep)
IPW
DM
Efficient
control
(true ea)
MSE
1.06
0.09
0.10
0.35
0.03
0.03
Bias
-0.03
0.19
0.18
-0.00
-0.01
-0.01
Cov. ratio
0,93
0.40
0.61
0.97
0.77
0.91
Table 3: Experimental results.
The upper and lower tables are results the censoring and case-
control settings, respectively.
IPW DM (estimated go)
Efficient
IPW
DM
Efficient
Censoring
(true Io)
MSE
5.03
0.23
0.13
1.25
0.07
0.09
Bias
-1.32
0.24
0.18
0.17
0.07
0.04
Cov. ratio
0.91
0.22
0.82
0.99
0.34
0.98
Case-
IPW
Efficient (estimated eo)
DM
IPW
DM
Efficient
control
(true ea)
MSE
0.40
0.03
0.03
0.23
0.01
0.01
Bias
-0.09
0.10
0.11
0.00
-0.02
-0.00
Cov. ratio
0.99
0.69
0.82
0.99
0.92
0.98
Censoring
IPW
DM
Efficient
Case-control
IPW
DM
Efficient
MSE
297.34
6.38
5.19
MSE
26.58
1.18
1.49
Bias
-16.36
-0.58
0.56
Bias
2.36
0.52
0.77
Cov. ratio
0.00
0.10
0.22
Cov. ratio
0.42
0.29
0.40
Table 4: Response surface A. Left: censoring setting; Right: case-control setting.
The observation indicator O is generated such that O = with probability if D; 1. and O = 0 D; 0, where is drawn from a uniform distribution over [0, 1] before the experiment begins. The outcome is generated Y (X 3)2 +.1+ To s, where (0 1) and set To 3.
The nuisance parameters are estimated using three-layer perceptrons with hidden layers of 100 nodes. The convergence rates satisfy Assumption 4.7 under standard conditions (Schmidt-Hicbe 2020).
We compare our proposed estimator, Prens-off with two alternative estimators: the IPW estimator Tn Goens-IPW and the DM estimator Pn 7 as defined in Remarks 4.4 and 4.4 respectively. Note that all of these estimators are proposed by and our objective is not to demonstrate that 7g outperforms the others, although we recommend propensity score is either estimated using the method proposed by Elkan & Noto (0088 assumed to be known.
Geeus-eff
practice. We consider both cases in which the
We set n = 3000. We conduct 5000 trials and report the empirical mean squared errors (MSEs) and biases for the true ATE, as well as the coverage ratio computed from the confidence intervals in Table 2 for = 3000 and Table 3 for = 500. We also present the empirical distributions of the ATE estimates in Figure for 3000 and Figure for 5000.
L.2 Case-control setting
In the case-control setting, covariates for the treatment and unknown groups generated from p 3-dimensional normal distributions: XT ~ (T, and (o(x) = co(1)CT,o(z) + (x)(((())) where (T.o( (x) and (o(x) are the densities of the normal distributions N (1, 4p) and N N( 1p Ip). respectively. We set up 0.5, pn = 0, and 1, = (11 ... 1)T. The class prior is set eo (1) = 0.3.
50

---

Estimates of the ATE (censoring setting)
IPW lestimated ge)
go!
Efficient (estimated
Empirical
IPW (tue dt)
Efficient true
Tue ATE
2.00
2.50
3.00
3.50
Estimates the ATE (case-control setting)
IPW/ (estimated
Empricc demsty
DM (estimated ga)
(estimated go)
(tree
Efficient true 0g)
2.00
2.25
2.50
2.75
3.00
3.25
330
3.75
4.00
ATE
Figure 3: Empirical distributions of ATE estimates.
Estimates ATE (censaring setting)
2.5
Empircel deraity
DM (estimated
2.0
Efficient testimated gol
1.5
(true
Efficient Itrue
0.5
2.25
2.50
3.25
4.00
Estimates the ATE (case-control setting)
IPW lestimated
Empincal dennsty
4
DM lestimated gv)
Efficent lestmateed go)
3-
gol
2-
(true
Bffccent trree ga)
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
Figure 4: Empirical distributions of ATE estimates.
Censoring
IPW
DM
Efficient
Case-control
IPW
DM
Efficient
MSE
327.49
4.15
1.14
MSE
46.15
3.34
3.77
Bias
-17.52
-1.58
-0.28
Bias
2.66
0.41
0.93
Cov. ratio
0.00
0.00
0.01
Cov. ratio
0.42
0.21
0.43
Table
Response surface Left: censoring setting; Right: case-control setting.
By definition, the propensity score co(d x:) is given by 6n(1 x)) co (z))))) The outcome is generated in the same manner in the censoring setting: Y (XT3)² +.1+ where Tp = 3.
Since we use neural networks, we estimate the propensity score go using the non-negative PU learn- ing method proposed by Kiryo et al. (2017), which is designed mitigate overfiuting when neural networks are applied. For simplicity, assume that the class prior co(1) is known.
51

---

Estimatis the ATE (censoring sctting)
Estimatis the AFE Jcase cantrol setting)
lntrmnnd
go)
instrnted pi
iustmaad
20C
Figure 5: Response surface A. Left: censoring setting, Right: case-control setting.
Estimates the ATE /oenacring weiing
Extirrtiin the AlE (case-control vetting)
tertmand pi
inttmate pJ
023
Instrated pd
020
6:0
0.5
222
222
Figure 6: Response surface B. Left: censoring setting; Right: case-control setting.
We consider two cases: (m,1) (1000, 2000) and (2000, 3000). We conduct 5000 trials and report the empirical mean squared errors (MSEs) and biases for the true ATE, along with the coverage ratio computed from the confidence intervals in Table for (m, () (1000, 2000) and Table for (m, 1) = (2000, 3000). We also present the empirical distributions of the ATE estimates Figure for (m, 1) (1000, 2000) and Figure (m. l) (2000, 3000).
M Empirical analysis using semi-synthetic data
In this section, investigate the empirical performance of our estimators using the Infant Health and Development Program (IHDP) dataset. The dataset contains simulated outcomes paired with covariates observed in the real world Hilll 20.
M.1 Dataset.
The sample size is 747, and the covariates consist of 6 continuous variables and 19 binary variables.
Hill (2011) considers two scenarios for the outcome models: response surface A and response sur- face B. Response surface A generates the potential outcomes Y;(1) and Y2(0) according to the fol-
lowing model:
(((0) ~ N(X, 7A, 7A,1),
Y((1) ~ ~XX 7A + 4,1),
where each element 74 E 2225 is randomly drawn from 1,,1,2,3,4) with probabilities (0.5, 0.2, 0.150,55,,. 0.1, 0.05).
In contrast, response surface B generates the potential outcomes Y2(1) and ((()) as follows:
Yı(0) ~ N (exp ((X, + W) TT)) 7B).1) .1),
Y((1) ~ (X. YB - 1),
where W is offset matrix of the same dimension X, with all elements equal 0.5, a constant chosen to normalize the average treatment effect conditional on d = to be 4, and each element of 7B E 1R25 is randomly drawn from 0.1 0.2, 0.3, 0.4) with probabilities (0.6, 0.1, 0.1, 0.1, 0.1,0,,,
M.2 Censoring setting
We first investigate the censoring setting. The other experimental setups are identical those in Section 6 Given {(X,, D,, Y)) from the IHDP dataset, we generate the observation indicator O
52

---

from a Bernoulli distribution with probability 0.1 if D, 1, and set O; = 0 D; = 0. The nuisance parameters are estimated using linear regression and (linear) logistic regression.
We compare our proposed estimator, goens-eft with two other candidates: the IPW estimator "n Ppens-IPW and the DM estimator Rx Preus-DM, defined in Remarks 44 and 4.4 respectively. All
of these estimators forms the others, although we recommend it in practice. Unlike in Section 6 we only consider the case in which the propensity score is estimated using the method proposed by Elkan & Noto (2008).
proposed by Our aim is not to demonstrate that Gcens-eff strictly outper- Tn
For each outcome model (response surface and B), we conduct 1000 trials and report the empirical mean squared errors (MSEs), biases for the true ATE, and the coverage ratio (Cov. ratio) computed from the confidence intervals Tables 4 and 5
We also present the empirical distributions of the
ATE estimates in Figures and
M.3 Case-control setting
In the case-control setting, we randomly split the dataset D into two subsets. One is used as an unlabeled dataset, and the other is used as positive dataset by selecting only the treated units from it. The class prior is set (1) 0.1.
For each outcome model (response surface and B), we conduct 1000 trials and report the empirical mean squared errors (MSEs), biases for the true ATE, and the coverage ratio (Cov. ratio) computed from the confidence intervals in Tables 4 and 5 We also present the empirical distributions of the ATE estimates in Figures 66
53


Reviews:
[{"review_id": "kDd9ooOClZ", "reviewer": "Reviewer_eyXp", "summary": "This paper studies a new setup of average treatment effect (ATE) estimation inspired by learning from positive and unlabeled data:\nwe have observations for which we know the treatment was given while we have other observations for which we do not know whether the treatment was given or not (i.e., in the control group). More specifically, the paper studies both censoring and case-control settings.\n\nFor each setting, the authors propose estimators based on the Inverse Probability Weighting (IPW), the Direct Method (DM), and the efficient (Eff) influence function.\nThe authors derive an explicit formula for the efficient influence function (Lemma 4.1) and show the efficiency bound (Theorem 4.2).\nThen, they propose the Eff estimator as the zero of the sample average of the influence function solved for the parameter of ATE (l.203).\nUnder several standard assumptions, they show their simplest form of double robustness result (Theorem 4.4).\n\nMoreover, assuming that the propensity score $g_0$ is known, they propose a cross-fitting-based estimator, for which they prove the asymptotic efficiency (Theorem 4.7) under the assumptions including those on some doubly-robust rates of convergence for the nuisance parameter estimates (Assumptions 4.6).\nThey also illustrate that the IPW and the DM estimators fail to achieve this asymptotic efficiency under those assumptions (Remarks in l.273 and l.282).\n\nThe authors then extend the result to the case where we need to estimate $g_0$, for which another dataset is assumed to be available.\nFor this methods, they show the asymptotic efficiency.\n\nFinally, the authors present simple simulation results confirming their theory.", "questions": "I will consider raising the score if these questions are addressed.\n\n### Major questions:\n- l.1300, I could not figure out how to obtain the first equality. Could the author give more details?\n- ll.155-156, \"the propensity scores $g_0(d | x)$ and $e_0(d | x)$ can be estimated using PU learning methods\"\nAre they identifiable? I thought we need to know the class prior in PU learning, and it requires some conditions for identifiability.\n- l.1250, what is $m$?\n- l.1260, where did $m$ come from? Where did $m$ and $\\frac{1}{\\bar{n}} \\sum_{i=1}{m}$ go after the equality?\n- l.1261--1262, how do I conclude that?\n\n### Minor questions:\n- l.1251, Is $C_{(\\ell)}$ defined? Is it $D_{(\\ell)}$?\n- l.1265, how did the authors deal with the random quantity $\\hat{\\pi}_n^{(\\ell)}$ in the denominator? A similar question applies to the last equality of l.1023.\n- l.1292, do we not need a boundedness of the functional for the Riesz representation theorem? (Do we really need the Riesz representation theorem if you anyway show Eq. (9) concretely?)\n- l.1298, did the authors use the fact that the scores have zero mean? Do we not need some regularity condition for this?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strengths\n- The setup is interesting and seems useful.\n- The theoretical results seem sound. I checked the proofs for the main results. (However, I have a few questions for clarification.)\n- The proposed method and the results seem highly non-trivial and relevant.\n\n### Weaknesses\n- There is no real data experiment.\n- Some parts of the proofs are too brief.\n- The notation of the math symbols is a bit confusing.", "was_revised": true, "final_justification": "During the discussion, we mainly discussed the following points:\n- Identification issue\n\nThe authors addressed this during the discussion. They provided references and explained when we have the identification result.\n\n- Clarification on the proofs\n\nThe authors addressed my concerns during the discussion. Some regularity conditions are missing, but the authors promised that they would add them. I am more convinced that the main theorems of the paper are correct.\n\n- No real-data experiment\n\nThis was not addressed in the discussion. However, I believe that the strong methodological and theoretical contributions compensate for this weakness.\n\n**My final evaluation:**\nThe paper works on an interesting problem, proposes a great method, and provides strong theoretical guarantees. Clarity issues have been addressed in the rebuttal. I would like to update my score from 4 to 5.\n\n-----\nMinor feedback:\nI think the proofs could be made more accessible by adding further details.", "rebuttal": "We thank you for your detailed and constructive feedback. We will correct typos the reviewer points out. Below, we address each of your points.\n\n-------\n\n**Q1**. l.1300, I could not figure out how to obtain the first equality. Could the author give more details?\n\n**A1**. We relied on the following from the first to the second equality:\n$$E[(1 - \\frac{e_0(1| X)}{e_0(0| X)})(Y(1) - \\mu_T(X))S_{X}(X; \\theta_0) | X] = (1 - \\frac{e_0(1| X)}{e_0(0| X)})S_{X}(X; \\theta_0)E[Y(1) - \\mu_T(X) \\mid X] = 0,$$\n\n$$E[\\frac{(Y_U - \\mu_{U, 0}(X)}{e_0(0| X)}S_X(X; \\theta_0) ] = 0, $$\n\n$$E[\\tau_0(S_X(X; \\theta_0) + S_{Y_U}(Y_U| X; \\theta_0)) ] = 0.$$\n\nWe missed a minus sign in front of $\\frac{Y_U - \\mu_{U, 0}(X)}{e_0(0| X)}$, which will be corrected in the next revision. The main ideas are\n- using $E[Y(1) - \\mu_{T, 0}(X)| X] = 0$ and $E[Y(0) - \\mu_{U, 0}(X)| X] = 0$,\n- recognising that the expectations of the score functions are zero.\nThese points will be clarified in the next revision.\n\n\n-------\n\n**Q2**. ll.155-156, \"the propensity scores $g_0(d\\mid x)$ and $e_0(d\\mid X)$ can be estimated using PU learning methods\" Are they identifiable? I thought we need to know the class prior in PU learning, and it requires some conditions for identifiability.\n\n**A2**. Identifiability differs between the censoring and case‑control settings and, in both, further depends on specific assumptions. Elkan and Noto (2006) show that under the censoring setting, the conditional class probability can be learned without the class prior, provided the SCAR assumption holds. Under the case‑control setting, some studies demonstrate that both the propensity score and the class prior can be identified simultaneously if parametric models are used for the propensity score (Lancaster and Imbens (1996) and Kato, Xu, Niu, and Sugiyama (2016)). There are also studies that investigate class prior estimation as an independent problem (Ramaswamy, Scott, and Tewari (2016) and du Plessis and Sugiyama (2016)).\n\nWe note that while PU learning with an unknown class prior has been rigorously addressed in the censoring setting, unresolved issues may remain in the case‑control setting. Although previous works, such as Lancaster and Imbens (1996) and du Plessis and Sugiyama (2016), have been well received and demonstrate empirical success, the conditions under which the class prior is identifiable have not been fully explored in detail, especially when using nonparametric propensity score models. Compared with standard logistic regression, PU learning introduces the class prior as an additional unknown. Therefore, we believe that the core issue lies in the number of unknown parameters and the information available in the data. From a data‑availability perspective, the censoring setting provides more information than the case‑control setting, which simplifies the problem.\n\nOur aim here is to present a straightforward solution under the most standard set of assumptions. We will clarify this point and add further references in the next revision.\n\n- Elkan and Noto (2008). \"Learning classifiers from only positive and unlabeled data.\"\n- Lancaster and Imbens (1996). \"Case-control studies with contaminated controls.\"\n- Kato, Xu, Niu, andSugiyama (2016). \"Alternate estimation of a classifier and the class-prior from positive and unlabeled data.\"\n- du Plessis and Sugiyama (2016). \"Class prior estimation from positive and unlabeled data.\"\n- Ramaswamy, Scott, and Tewari (2016). \"Mixture proportion estimation via kernel embeddings of distributions\"\n\n-------\n\n**Q3a**. l.1250, what is $m$?  \n**Q3b**. l.1260, where did $m$ come from? Where did $m$\n and $\\frac{1}{\\overline{n}}\\sum^m_{i=1} go after the equality?\n\n**A3**. Thank you for pointing out the typo. Both instances of $m$ should be $\\overline{n}$. We will fix this in the next revision.\n\n-------\n\n**Q4**. l.1261--1262, how do I conclude that?\n\n**A4**. As the reviewer notes in Q6 and Q7, the result follows from combining the boundedness assumption with the consistency of each nuisance‑parameter estimator. We will clarify this point in the next revision.\n\n\n-------\n\n**Q5**. l.1251, Is $C_{(\\ell)}$ defined? Is it $D_{(\\ell)}$?\n\n**A5**. This is indeed a typo; it should be $\\mathcal{D}^{(\\ell)}$. We will amend it in the next revision.\n\n\n-------\n \n**Q6**. l.1265, how did the authors deal with the random quantity \n$\\hat{\\pi}_n^{(\\ell)}$ in the denominator? A similar question applies to the last equality of l.1023.\n\n**A6**. We appreciate the reviewer's comment. In l.1265 and l.1023, boundedness of the estimators is required. We had omitted these boundedness assumptions and will add them in the next revision.\n\n\n-------\n \n**Q7**. l.1292, do we not need a boundedness of the functional for the Riesz representation theorem? (Do we really need the Riesz representation theorem if you anyway show Eq. (9) concretely?)\n\n**A7**. The boundedness of the functional is indeed necessary for the Riesz representation theorem. More concretely, we ned to assume that the score function needs to belong the L2 space. In the next revision, we will add assumptions about the boundedness of the conditional means and conditional variances of the outcomes.\n\nThe Reisz representation theorem itself is also required to guarantee the existence of an efficient influence function. However, as the reviewer commented, if you anyway show Eq. (9) concretely, we can pass the use of the Riesz representation theorem at least in our lemma. Note that we finally need to use the theorem to discuss the lower bound of the asymptotic variance, for which we only cited van der Vaart (1998). We will explicitly assume boundedness of the score functions in l.1285 in the next revision. \n\n-------\n\n**Q8**. l.1298, did the authors use the fact that the scores have zero mean? Do we not need some regularity condition for this?\n\n**A8**. We used the property explained in A1 of this reply. Regularity conditions are not required once we can exhibit a specific score function and verify that it satisfies the definition for this problem."}, {"review_id": "4NzFb5KC8U", "reviewer": "Reviewer_HcTq", "summary": "This work studies the problem of estimating the average treatment effect in a setting analogous to the positive and unlabeled setting that has been previously studied in the supervised learning literature. Specifically, there is a set of known treated units and a second set of units for which the treatment status is unknown. The authors consider two variants of this problem, in the case-control setting the two groups are independent groups, in the censored setting there is a random process that dictates membership to the positive and unlabeled groups. The authors develop a semi-parametric estimator and derive the influence functions and asymptotic normality for both of these settings. In each, the authors leverage PU estimators from the literature. The propensity score amongst the unlabeled units is assume known in the main text for the proof of asymptotic normality. This is relaxed later in the paper by assuming the availability of an auxiliary dataset.", "questions": "* There is a fairly large set of methods that have been proposed for PU learning. It would be helpful if the authors could provide some intuition here for how the choice of PU learning method affects the proposed method, e.g., employing Elkon, vs du Plessis et al. \n\n* Can the authors provide some intuition on the effect of having to estimate the g function on the overall performance of the estimator? As I mentioned above, it seems like there would be an interaction between that and the results both simulation and theoretical presented, but it's not entirely clear to me from reading.", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n* The problem studied in this paper is interesting, and novel to the causal setting. The authors do a nice job of making the connection between PU learning and semi-parametric estimation. \n* The derivation of $\\sqrt{n}$ consistency and asymptotic normality is nice to see. \n* The authors do a nice job of centering the proposed approach with the derivation of the IF. The assumptions are clearly laid out, and well stated throughout.\n* The proposed approach shows good empirical performance against sensible alternatives. \n\nWeaknesses: \n* The largest weakness with this paper is the assumption that the g function is known throughout. The authors do present a solution to this by having an auxiliary data set. However, it's not entirely clear to me how this will affect the results. My intuition is that the convergence rate is now going to be dominated by the smaller of the two data sets, either the one used to estimate the g function or the one used to estimate the rest of the estimator. \n\n* It would be really useful if the authors could contextualize the results and presented methodology within the larger framework of missing data. They do a nice job of setting up the missingness mechanism, SCAR. However, I think the paper would have greater impact if there was some discussion of this method and its relation to other approaches for missing data, such as the Heckman model or, e.g.Robins, Rotnitzky, and Zhao (1994). I do recognize that the proposed setting is unique, however I just think it would help gain intuition for the reader.", "was_revised": true, "final_justification": "I would like to thank the authors for their clarification. In my view, my concerns have been well addressed and I have updated my score to reflect this. This paper provides a nice addition to the literature, addressing a problem of practical importance that, at least to my knowledge, has not been studied in the literature to this point.", "rebuttal": "We appreciate the reviewer’s insightful comments. Below, we respond to each of the points raised.\n\n------\n\n**Q1**. \nThere is a fairly large set of methods that have been proposed for PU learning. It would be helpful if the authors could provide some intuition here for how the choice of PU learning method affects the proposed method, e.g., employing Elkon, vs du Plessis et al.\n\n**A1**. We appreciate your thoughtful suggestion. The choice of PU learning method should reflect the underlying data‑generating process. For instance, Elkan and Noto (2008) and du Plessis et al. (2015) tackle essentially different problems: the former studies the censoring setting, whereas the latter considers the case‑control setting. Under the censoring setting we can estimate the conditional class probability, that is, the propensity score, without knowing the class prior and under weaker assumptions than in the case‑control setting.\n\nOther methods, such as Bekker and Davis (2018) and Kato, Teshima, and Honda (2019), address selection‑bias issues in PU learning. When using neural networks, Kiryo, Niu, du Plessis, and Sugiyama (2017) propose a non‑negative correction that mitigates overfitting bias. Many approaches, especially for censoring PU learning, are summarized in the survey by Bekker and Davis (2020).\n\nFor the class prior, for example, Lancaster and Imbens (1996) discuss that it can be estimated simultaneously with the conditional class probability via the generalized method of moments, provided parametric models are used for the propensity score. Ramaswamy, Scott, and Tewari (2016), as well as du Plessis and Sugiyama (2016), study class prior estimation as an independent problem of interest.\n\nWe currently explain these methods in the main body and Appendices A and B. In the next revision, we will explain them in more details.\n\n- Elkan and Noto (2008). \"Learning classifiers from only positive and unlabeled data.\"\n- du Plessis, Gang. Niu, and Masashi Sugiyama (2015). \"Convex formulation for learn395 ing from positive and unlabeled data.\"\n- du Plessis and Sugiyama (2014). \"Class prior estimation from positive and unlabeled data.\"\n- Lancaster and Imbens (1996) \"Case-control studies with contaminated controls.\" Journal of Econometrics.\n- Bekker and Davis (2018). \"Learning from positive and unlabeled data under the selected at random assumption.\"\n- Bekker and Davis (2020). \"Learning from positive and unlabeled data: a survey.\"\n\n------\n\n**Q2a.** The largest weakness with this paper is the assumption that the g function is known throughout. The authors do present a solution to this by having an auxiliary data set. However, it's not entirely clear to me how this will affect the results. My intuition is that the convergence rate is now going to be dominated by the smaller of the two data sets, either the one used to estimate the g function or the one used to estimate the rest of the estimator.  \n**Q2b**. Can the authors provide some intuition on the effect of having to estimate the g function on the overall performance of the estimator? As I mentioned above, it seems like there would be an interaction between that and the results both simulation and theoretical presented, but it's not entirely clear to me from reading.  \n\n**A2.** Thank you for your comments. First, we clarify that the convergence rate is not dominated by the smaller of the two data sets. In estimating the $g$ function, we can use the entire set of data, including both the additional dataset and the original dataset used for estimating the other parameters. As we stated in our draft, \"We can also use $\\{(X_i, O_i)\\}^n_{i=1}$ from $\\mathcal{D}$ to estimate $g_0$ with $\\mathcal{D}^{aux}$. The inclusion of $\\{(X_i, O_i)\\}^n_{i=1}$ can improve empirical performance.\" The key point is that the $g$ function must be estimated at a faster rate than the other parameters; therefore, we use both the original and the additional datasets. This also reflects a difference in asymptotic frameworks. In our proof, we take the limit with $n^{aux} \\to \\infty$ first, followed by $n \\to \\infty$. Alternatively, one could consider an asymptotic regime where both $m$ and $n$ go to infinity simultaneously (i.e., $n^{aux}, n \\to \\infty$). In the latter case, it is required that $m$ diverges at a faster rate than $n$.\n\nAdditionally, we consider the following points regarding the $g$ function:\n- We need to impose additional assumptions or use additional datasets to theoretically obtain asymptotic normality.\n- We can prove consistency without using additional datasets, but there is no double robustness between the propensity score and the observation probability.\n- The empirical performance depends on the correctness of the model specification or the datasets used.\nIn the next revision, we will explain more clearly why the $g$ function must be known in order to establish asymptotic normality. \n\nTo support the above points, we outline the mechanism why we need the true $g$ below and also refer to our reply **A5** to Reviewer D1gL. \n\nWe begin by noting the absence of a doubly robust structure linking the propensity score with the other nuisance parameters (the conditional expected outcome and the observation probability) in our setting. It is important to emphasize that double robustness is a key technical tool for establishing asymptotic normality, not only consistency. Double robustness is used as an analytical device derived from the efficiency bound and is often justified via the canonical gradient or Neyman orthogonality (see Schuler and van der Laan, 2024). This structure makes the ATE estimator insensitive to the estimation of the nuisance parameters in the standard ATE estimation setting. Essentially, it forms a product of the estimation errors of the nuisance parameter estimators, which reduces the overall estimation error more quickly.\n\nIn our framework, double robustness holds between the observation probability and the conditional expected outcome: given a consistent estimator of the propensity score, if either the observation probability or the conditional expected outcome is consistently estimated, we obtain a consistent ATE estimator. In contrast, in standard ATE estimation, double robustness holds between the propensity score and the conditional expected outcome: if either of those two components is consistently estimated, the ATE is also consistently estimated.\n\nThe lack of double robustness in PUATE can be explained as follows. Estimating $E[Y(0)\\mid X]$ requires the propensity score itself, whereas in the standard case, the two nuisance parameters can be estimated independently. Concretely, we use\n$$ E[Y(0) \\mid X] = \\left(E[Y \\mid X] - g_0(0 \\mid X) E[Y(1) \\mid X]\\right) / g_0(1 \\mid X). $$\nBecause of this dependence, we cannot obtain the doubly robust structure. Since we cannot obtain the doubly robust structure, a bias‑reducing product term such as $\\lVert g_0 - \\hat{g} \\rVert \\lVert \\nu_0 - \\hat{\\nu} \\rVert$ also does not appear. As shown in Lemma 4.1, the efficient influence function therefore lacks a doubly robust structure, and standard proofs based on Neyman orthogonality cannot deliver asymptotic normality without stronger conditions. An auxiliary dataset that enables a sufficiently fast first‑stage estimate of the propensity score is a practical solution.\n\n- Alejandro Schuler and Mark van der Laan. *Introduction to Modern Causal Inference*, 2024\n\n------\n\n\n**Q3**. It would be really useful if the authors could contextualize the results and presented methodology within the larger framework of missing data. They do a nice job of setting up the missingness mechanism, SCAR. However, I think the paper would have greater impact if there was some discussion of this method and its relation to other approaches for missing data, such as the Heckman model or, e.g. Robins, Rotnitzky, and Zhao (1994). I do recognize that the proposed setting is unique, however I just think it would help gain intuition for the reader.\n\n**A3**. Thank you for the suggestion. We currently frame our results within causal inference under stratified sampling. Another example of stratified sampling is covariate shift; Uehara, Kato, and Yasui (2021) study causal inference when the training data contain (covariates, treatment, outcomes) but the evaluation data contain only covariates. Concerning identifiability in PU learning, in addition to the studies cited in A1 (Bekker and Davis (2018) and Kato, Teshima, and Honda (2019)) the classical work of Lancaster and Imbens (1996) is also relevant, as answered in **A1**. \n\nThe Heckman model and the framework of Robins, Rotnitzky, and Zhao (1994) are important contributions to causal inference with missing data. The Heckman model addresses endogeneity between the outcome and the observation indicator, whereas our analysis does not consider such endogeneity. Thus our setting is simpler in that respect, yet it is more challenging because the relationship between treatment assignment and observation follows the PU learning mechanism. Similarly, Kennedy (2020) allows for endogeneity between treatment observation and outcome, but the observation mechanism itself is simpler than in our work. These approaches are not nested within one another; rather, they are complementary, and combining them may yield more flexible statistical modelling.\n\nWe summarize some of these related studies in Appendices A and B and intend to provide additional details in the camera ready version.\n\n- Uehara, Kato, and Yasui (2021). \"Off-policy evaluation and learning for exter516 nal validity under a covariate shift.\"\n- Kennedy (2020). \"Efficient nonparametric causal inference with missing exposure information.\"\nThe International Journal of Biostatistics."}, {"review_id": "X5NAVlRg60", "reviewer": "Reviewer_D1gL", "summary": "This paper investigates treatment effect estimation in settings where treatment assignments are missing for some observations, including both censoring and case-control scenarios. The authors establish identification and semiparametric efficiency theory, and demonstrate the performance of the proposed doubly robust estimators through theoretical analysis and empirical results.", "questions": "1. The significance of the work could be enhanced by incorporating additional application scenarios and concrete examples that align with the settings considered in the paper. This would better demonstrate the practical utility of the proposed methods.\n2. In Assumption 3.1, the authors should consistently use “o” as the realization of the variable \"O” rather than switching to “d,” which is used as the realization of “D.” A similar inconsistency appears in Assumption 3.3. It is recommended that the authors carefully review the manuscript and ensure consistent use of notation throughout to improve clarity.\n3. Assumption 3.1 posits that the missingness indicator O is independent of X for treated units, which corresponds to a Missing Completely At Random (MCAR) assumption. This is a strong assumption in many practical scenarios. It would be helpful for the authors to discuss why this assumption is necessary in the current framework and to explore whether a relaxation to a Missing At Random (MAR) assumption could be considered.\n4. Section 3 presents an IPW-style identification formula for the average treatment effect (ATE). It would be valuable to also include an identification formula based on outcome regression, such as $$E[Y(1)]=E[E(Y \\mid X, A=1)]$$ as commonly used in standard causal inference settings. Providing a derivation of the identification results—perhaps in an appendix—would also help readers better understand how the assumptions in Section 3 enable identification.\n5. The main result (Theorem 4.7) is currently stated under the assumption that the propensity score is known. Section 4.5 shows that the same result holds when the propensity score is estimated from an independent sample. It may improve the flow and clarity to directly present the main results under the sample-splitting framework (e.g., divide the sample into L+1 folds with one fold training the propensity score), which is widely adopted in practice. As long as the number of folds is fixed, the final convergence rates remain valid.", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Overall, this paper addresses an important and interesting problem in causal inference. The proposed methods are supported by solid theoretical guarantees and empirical evidence, and the presentation is clear and well-organized. The manuscript would benefit from a deeper discussion of the practical applications and real-world relevance of the proposed methods. More detailed comments on specific aspects are provided below.\nQuality: The theoretical foundations and proposed methods are solid. I have listed a few questions regarding them in the 'Questions' section below.\nClarity: The paper is clearly written and well organized. To maintain clarity and avoid excessive length, the details for the case-control setting are appropriately moved to the appendix. I have also included some comments on improving the notation in the 'Questions' section below.\nSignificance: While the theoretical development is solid, the authors should place greater emphasis on the practical applicability of their setting and methods.\nOriginality: The results in this paper are novel to the best of my knowledge.", "was_revised": true, "final_justification": "This paper presents an interesting and well-developed framework, and I believe it can serve as a valuable contribution to the literature.", "rebuttal": "We appreciate the reviewer's detailed comments. Below, we address each of the points raised in the review. \n\n--------\n\n**Q1**. The significance of the work could be enhanced by incorporating additional application scenarios and concrete examples that align with the settings considered in the paper. This would better demonstrate the practical utility of the proposed methods.\n\n**A1**. Thank you for the suggestion. In the next revision, we will incorporate the following additional application scenarios and concrete examples that match the settings of our study. \n- We are interested in how building an online store affects product sales. In a recommendation system, a customer who purchases a product through the website is known to have visited the site, whereas a customer who buys the product in a physical store may or may not have visited the website. When the construction of the online store is regarded as the treatment, customers who purchase in‑store belong to the unknown group.\n- In a similar vein, consider the effect of distributing coupons on product sales. Because the logging system is imperfect, we observe two datasets: purchase records for customers who used coupons, and purchase records that mix customers who did and did not use coupons. Here, customers who appear only in the mixed dataset constitute the unknown group.\n- A company unintentionally sold defective products, for example, vehicles with faults or food containing impurities. Consumers who notice the defect report both the defect and any resulting damage, such as health issues or accidents, to the company. Consumers who do not notice the defect file no report. In this setting, consumers who report the defect form the treatment group, while those who do not notice the defect belong to the unknown group.\n\nOur method is also useful when the treatment can be defined only by contrasting it with a non‑treatment condition. For example, consider examining how excessive work hours affect workers' mental health. Courts and labor inspectors can identify illegal overtime, but it is difficult to decide whether other working hours are excessive. Firms found to have engaged in illegal overtime can therefore serve as the treatment group, whereas all other firms constitute the unknown group. Similarly, by taking units with easily observable outcomes as the treatment group and all others as the unknown group, one can define the comparison group simply as not treated. This idea is widely adopted in anomaly‑detection approaches that rely on density‑ratio or PU learning.\n\n\n--------\n\n**Q2**. In Assumption 3.1, the authors should consistently use “o” as the realization of the variable $O$ rather than switching to “d,” which is used as the realization of $D$. A similar inconsistency appears in Assumption 3.3. It is recommended that the authors carefully review the manuscript and ensure consistent use of notation throughout to improve clarity.\n\n**A2**. We appreciate the reviewer’s comment. This is a typographical error, and we will correct it in the next revision.\n\n--------\n\n**Q3**. Assumption 3.1 posits that the missingness indicator $O$ is independent of $X$ for treated units, which corresponds to a Missing Completely At Random (MCAR) assumption. This is a strong assumption in many practical scenarios. It would be helpful for the authors to discuss why this assumption is necessary in the current framework and to explore whether a relaxation to a Missing At Random (MAR) assumption could be considered.\n\n**A3**. This assumption is introduced by Elkan and Noto (2008) as an analogue of the MCAR assumption and is widely accepted as a baseline in the PU learning literature. As explained in our draft (line 150), several methods have already been proposed for PU learning without the SCAR assumption. Because our paper does not focus on propensity score estimation, we present the analysis under the standard setting.\n\n In the censoring PU learning setting, Bekker and Davis (2018) demonstrates possible relaxations, while in the case‑control setting Kato, Teshima, and Honda (2019) offers related results. We will add a fuller discussion and more references in the next revision.Another key issue is the need to estimate the class prior, which is an additional unknown parameter relative to standard logistic regression. Identifying this parameter requires assumptions.\n\n- Bekker and Davis (2018). \"Learning from positive and unlabeled data under the selected at random assumption.\"\n- Kato, Teshima, and Honda (2019). \"Learning from positive and unlabeled data with a selection bias.\"\n\n\n**Q4**. Section 3 presents an IPW-style identification formula for the average treatment effect (ATE). It would be valuable to also include an identification formula based on outcome regression, such as\n$$E[Y(1)] = E[E[Y | X, A = 1]]$$\nas commonly used in standard causal inference settings. Providing a derivation of the identification results—perhaps in an appendix—would also help readers better understand how the assumptions in Section 3 enable identification.\n\n**A4**. In the censoring setting, an outcome regression‑based identification formula is indeed feasible. In the case‑control setting, however, treatment is deterministic for the treated group, so conditioning on $A = 1$ is not directly appropriate. We can still define $p(y \\mid x, a = 1)$, but the interpretation differs. We will clarify these points and, where possible, supply derivations in an appendix.\n\n--------\n\n**Q5**. The main result (Theorem 4.7) is currently stated under the assumption that the propensity score is known. Section 4.5 shows that the same result holds when the propensity score is estimated from an independent sample. It may improve the flow and clarity to directly present the main results under the sample‑splitting framework (e.g., divide the sample into $L+1$ folds with one fold training the propensity score), which is widely adopted in practice. As long as the number of folds is fixed, the final convergence rates remain valid.\n\n**A5**. Thank you for the suggestion. We consider that sample splitting alone does not resolve the issue, because the bottleneck lies in the convergence rate for the sample size $n$. Even with sample splitting, this rate cannot be improved.\n\nWhen constructing semiparametric efficient estimators, two ingredients are usually relied upon, namely a doubly robust structure and control of function complexity. In our setting, the doubly robust structure involves the observation probability and the conditional expected outcome, but not the propensity score. Specifically, conditional on a consistent estimator of the propensity score, if either the observation probability or the conditional expected outcome is consistently estimated, we can estimate the ATE consistently.\n\nThe doubly robust structure contributes not only to consistency but also to rapid bias reduction, a property often referred to as Neyman orthogonality. To illustrate this, consider a toy example in which our goal is to estimate an unknown parameter $\\theta$ under a data‑generating process that depends on two unknown functions $f$ and $g$ (not $g$ in our paper). For instance, $\\theta$ may represent the ATE, and $f$ and $g$ correspond to the conditional expected outcome and the propensity score, respectively. If we obtain a doubly robust estimator for $\\theta$, we can bound the bias of $\\sqrt{n}(\\theta - \\hat{\\theta}_n)$ by the product $\\|| f - \\hat{f}_n \\||_2 \\|| g - \\hat{g}_n \\||_2$, where $\\hat{f}_n$ and $\\hat{g}_n$ are estimators of $f$ and $g$, respectively.This product structure enables $\\sqrt{n}$‑consistency. While $\\|| f_0 - \\hat{f}_n \\||_2$ and $\\|| g_0 - \\hat{g}_n \\||_2$ individually may not reach the $\\sqrt{n}$ rate, for example, nonparametric estimators often converge at slower rates such as $n^{-1/3}$ or $n^{-1/4}$ depending on smoothness, their product can converge faster than $\\sqrt{n}$. This is the key advantage of the doubly robust structure for bias reduction.\n\nIn PUATE, however, although a doubly robust structure exists between the observation probability and the conditional expected outcome, it does not arise between the propensity score and the remaining nuisance parameters. Consequently, the estimation error of the propensity score may persist. Specifically,  \n\n$$\\sqrt{n}(\\hat{\\tau) - \\tau) = G + Bias1 + Bias2,$$\n\nwhere $G$ is asymptotically normal, Bias1 denotes the product of the estimation errors for the observation probability and the conditional expected outcome, and Bias2 denotes the estimation error of the propensity score. Bias1 can be $o_p(n^{-1/2})$, but Bias2 is not $o_p(n^{-1/2})$ unless the propensity score is estimated at a rate faster than the standard parametric rate $O_p(n^{-1/2})$. This is why our analysis assumes that the propensity score is known, or is estimated more rapidly using an independent dataset. When using such an independent dataset, in our proof, we take the limit with $m \\to \\infty$ first, followed by $n \\to \\infty$. Alternatively, one could consider an asymptotic regime where both $n^{aux}$ and $n$ go to infinity simultaneously (i.e., $n^{aux}, n \\to \\infty$). In the latter case, it is required that $m$ diverges at a faster rate than $n$."}, {"review_id": "UsndVZbJag", "reviewer": "Reviewer_TbdE", "summary": "This paper proposes a novel framework for estimating average treatment effects (ATE) using only treated and unlabeled data. It derives semiparametric efficiency bounds for two common missing data settings (censoring and case-control) and proposes estimators that are provably efficient. Theoretical results are supported by simulations showing improved performance over standard methods like inverse probability weighting.", "questions": "- Asymptotic normality in this paper requires the *true* propensity score (or an auxiliary dataset), whereas standard doubly-robust AIPW only needs an estimated propensity score converging at a fast enough rate. What structural aspect of the treated-plus-unlabeled setup makes propensity-score error first-order, and do you see other ways to recover asymptotic normality using only the original data (i.e., without bringing in a separate dataset)?\n- Could you provide a concrete, real-world example where only treated and unlabeled units are observed, and where ATE is the relevant causal estimand?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths**:\n\n- The claims made in the paper are well supported by both theoretical derivations and simulation studies\n- The writing is very clear and the contributions are well connected to the broader literature on treatment effect estimation (I appreciated the extended discussion in Appendix A)\n\n\n\n**Weaknesses**: \n\n-  The introduction does not clearly justify when and why the treated-plus-unlabeled scenario arises naturally in causal inference, beyond a brief and somewhat unclear recommendation-system example. It is not obvious to me how common this setting is in real-world causal inference\n- The censoring setting rests on the “selected completely at random” assumption that observation probability is constant within the treated class, an analogue of MCAR that is rarely satisfied", "was_revised": true, "final_justification": "I am maintaining my original score. The authors provided a clear explanation for the lack of double robustness in their setting and justified the need for stronger assumptions to achieve asymptotic normality. They also addressed my concern about real-world relevance with several concrete examples.", "rebuttal": "We would like to thank the reviewer for the detailed feedback. Below, we address each point raised by the reviewer.\n\n------\n\n**Q1**. Asymptotic normality in this paper requires the true propensity score (or an auxiliary dataset), whereas standard doubly robust AIPW only needs an estimated propensity score converging at a fast enough rate. What structural aspect of the treated‑plus‑unlabeled setup makes propensity‑score error first‑order, and do you see other ways to recover asymptotic normality using only the original data (i.e., without bringing in a separate dataset)?\n\n**A1**. We believe that in our case, when the propensity score is unknown, guaranteeing asymptotic normality requires additional assumptions, such as access to an auxiliary dataset for estimating the score. The mechanism is as follows.\n\nDouble robustness is a key technical tool for establishing asymptotic normality, not only for consistency. However, we note that in our setting, such a doubly robust structure does not hold between the propensity score and the other nuisance parameters (the conditional expected outcome and the observation probability). In our framework, double robustness holds between the observation probability and the conditional expected outcome: given a consistent estimator of the propensity score, if either the observation probability or the conditional expected outcome is consistently estimated, we obtain a consistent ATE estimator.\n\nNote that under the standard setup of ATE estimation, double robustness instead holds between the propensity score and the conditional expected outcome: if either of those two components is consistently estimated, the ATE is consistently estimated. This difference between the PUATE and the standard ATE frameworks stems from the identification of the expected outcomes of the control group. In PUATE, estimating $E[Y(0)\\mid X]$ requires the propensity score itself, whereas in the standard case the two nuisance parameters can be estimated independently. Concretely, we use\n$$ E[Y(0) | X] = (E[Y| X] - g_0(0 | X) E[Y(1) | X])/g_0(1 | X). $$\n\nIn the standard ATE estimation, $\\sqrt{n}$-consistency of the doubly robust estimator comes from the product of the estimation errors of nuisance parameters. However, because of the above dependence between the propensity score and the expected outcome, a bias‑reducing product term such as $\\|| g_0 - \\hat g\\||\\|| \\nu_0 - \\hat\\nu\\||$ does not appear. We can confirm this property from Lemma 4.1. As shown in Lemma 4.1, the efficient influence function therefore lacks such a doubly robust structure.\n\nThus, we lack the doubly robust structure that is usually required for $\\sqrt{n}$-consistency. Consequently, the estimation error of the propensity score may persist. Specifically,  \n\n$$\\sqrt{n}(\\hat{\\tau) - \\tau) = G + Bias1 + Bias2,$$\n\nwhere $G$ is asymptotically normal, Bias1 denotes the product of the estimation errors for the observation probability and the conditional expected outcome, and Bias2 denotes the estimation error of the propensity score. Bias1 can be $o_p(n^{-1/2})$, but Bias2 is not $o_p(n^{-1/2})$ unless the propensity score is estimated at a rate faster than the standard parametric rate $O_p(n^{-1/2})$. This is why our analysis assumes that the propensity score is known, or is estimated more rapidly using an independent dataset.\n\nTherefore, without stronger conditions, we cannot establish asymptotic normality in our setting. For example, an auxiliary dataset that enables a sufficiently fast first‑stage estimate of the propensity score is a practical solution.\n\nNote that double robustness is an analytical device derived from the efficiency bound and is often formalized via the canonical gradient or Neyman orthogonality (see Schuler and van der Laan, 2021). We also explain this problem in the reply A5 to Reviewer D1gL. We will explain this problem more clearly in the next revision.\n\n- Alejandro Schuler and Mark van der Laan. Introduction to modern causal inference, 2021. \n\n------\n\n**Q2**. Could you provide a concrete, real‑world example where only treated and unlabeled units are observed, and where ATE is the relevant causal estimand?\n\n**A2**. Thank you for the suggestion. In the next revision, as we also mentioned in our reply to Reviewer D1gL, we will incorporate the following application scenarios and concrete examples that match the setting of our study.\n\n- We are interested in how opening an online store affects product sales. Within a recommendation system, a customer who purchases a product through the website is known to have visited the site, whereas a customer who buys the product in a physical store may or may not have visited the website. Treating the launch of the online store as the treatment, customers who purchase in store constitute the unknown group.\n\n- In a similar vein, consider the effect of distributing coupons on product sales. Because the logging system is imperfect, we observe two data sets: purchase records for customers who used coupons and purchase records that mix customers who did and did not use coupons. Customers who appear only in the mixed data set form the unknown group.\n\n- A company unintentionally sells defective products, for example vehicles with faults or food containing impurities. Consumers who notice the defect report both the defect and any resulting damage, such as health issues or accidents, to the company. Consumers who do not notice the defect file no report. In this setting, consumers who report the defect form the treatment group, whereas those who do not belong to the unknown group.\n\nOur method is also useful when the treatment can be defined only by contrasting it with a non‑treatment condition. For example, consider examining how excessive work hours affect workers' mental health. Courts and labor inspectors can identify illegal overtime, but it is difficult to decide whether the other companies do not impose such overworking to employees. In this case, firms found to have engaged in illegal overtime can therefore serve as the treatment group, whereas all other firms constitute the unknown group. Then, we identify the treated units from the unknown group by using our method.  Similarly, by taking units with easily observable outcomes as the treatment group and all others as the unknown group, one can define the comparison group simply as control. This idea is widely adopted in anomaly‑detection approaches that rely on density‑ratio or PU learning.\n\n- Hido, Tsuboi, Kashima, Sugiyama, and Kanamori (2008). \"Inlier-Based Outlier Detection via Direct Density Ratio Estimation.\""}]

Author rebuttal:

Meta-review: This paper presents a novel and valuable contribution to causal inference by developing semiparametric efficient estimators for the Average Treatment Effect (ATE) in a Positive and Unlabeled (PU) learning setting. The reviewers unanimously found the problem interesting. After a thorough discussion period where the authors provided detailed and constructive rebuttals, all reviewers converged on a positive recommendation for acceptance. I agree with the reviewers and recommend acceptance. 

For the camera-ready version, the authors should incorporate the following revisions: 
- enhance the clarity and completeness of all theoretical results by adding omitted proofs, missing conditions and making the proofs and technical results more accessible;
- correct all identified typos and notational inconsistencies throughout the manuscript to improve readability;
- strengthen motivation and context by Integrating real-world examples and expanding the discussion to better contextualize the work within the broader literature on causal inference with missing data.
Decision: Accept (poster)