Title: Estimation of Treatment Effects in Extreme and Unobserved Data
Venue: NeurIPS 2025
Accepted: yes
Keywords: Causal Inference

Extreme Value Theory

Treatment Effect
Abstract: Causal effect estimation seeks to determine the impact of an intervention from observational data. However, the existing causal inference literature primarily addresses treatment effects on frequently occurring events. But what if we are interested in estimating the effects of a policy intervention whose benefits, while potentially important, can only be observed and measured in rare yet impactful events, such as extreme climate events? The standard causal inference methodology is not designed for this type of inference since the events of interest may be scarce in the observed data and some degree of extrapolation is necessary. Extreme Value Theory (EVT) provides methodologies for analyzing statistical phenomena in such extreme regimes. We introduce a novel framework for assessing treatment effects in extreme data to capture the causal effect at the occurrence of rare events of interest. In particular, we employ the theory of multivariate regular variation to model extremities. We develop a consistent estimator for extreme treatment effects and present a rigorous non-asymptotic analysis of its performance. We illustrate the performance of our estimator using both synthetic and semi-synthetic data.

Estimation of Treatment Effects in Extreme and Unobserved Data

Abstract

Causal effect estimation seeks to determine the impact of an intervention from observational data. However, the existing causal inference literature primarily addresses treatment effects on frequently occurring events. But what if we are interested in estimating the effects of a policy intervention whose benefits, while potentially important, can only be observed and measured in rare yet impactful events, such as extreme climate events? The standard causal inference methodology is not designed for this type of inference since the events of interest may be scarce in the observed data and some degree of extrapolation is necessary. Extreme Value Theory (EVT) provides methodologies for analyzing statistical phenomena in such extreme regimes. We introduce a novel framework for assessing treatment effects in extreme data to capture the causal effect at the occurrence of rare events of interest. In particular, we employ the theory of multivariate regular variation to model extremities. We develop a consistent estimator for extreme treatment effects and present a rigorous non-asymptotic analysis of its performance. We illustrate the performance of our estimator using both synthetic and semi-synthetic data.

1 Introduction

We are interested in studying the effect of treatment e.g., different policies and drugs, on rare yet impactful events such as large wildfires, hurricanes, tsunamis and climate change. These kinds of events happen at an extremely low frequency, but they can cause considerable damage to properties and pose serious threats to people's lives. For instance, we may want to know the effect of more

infrastructure investment or other kinds of precautionary policies on earthquakes. In many applications - from financial risk to environmental policy - it isn't enough to know how a treatment changes the average outcome; decision-makers care about whether it alters the extreme tail. More formally, we may want to estimate the effect of treatment D on outcome Y , conditioning on some extreme events. Estimating this kind of effect can help policymakers evaluate the impact of a policy and choose the best policy to reduce economic loss and save more lives when disasters happen.

Despite its clear importance, existing methods fall into two largely disconnected strands, each of which cannot fully address this question. One approach comes from the causal inference literature. Causal inference provides a comprehensive framework for counterfactual reasoning. Causal effect estimation is an important problem in this area, which finds wide applications in healthcare, education, business decision-making, and policy evaluation. Classic causal inference literature mainly focuses on estimating the average effects among certain groups. Little attention is paid to the causal effect on rare events. The scarcity of extreme data makes inference more challenging than in classic settings. As a result, naively applying classic causal inference estimation methods will produce poor results with large statistical error. For example, when making policies about earthquakes, we are usually unable to see a strong signal from historical data, as large earthquakes rarely occur and there are fewer samples in the dataset.

On the other hand, the Extreme Value Theory (EVT) studies the tail behaviors for statistical distributions, which provides the ideal tools for analyzing rare events. However, this approach does not take the data structure into consideration. In particular, it does not accommodate counterfactual treatments or adjust for covariates, so it cannot tell us what would happen under an intervention.

To bridge these gaps, we combine causal inference theory with EVT to provide a novel framework for extreme effect measurement. Following researches in EVT Coles et al. [2001], we use a multivariate regularly varying variable U to model extreme events. The rare event can be modeled by the event {∥ U ∥ > t } for large t . Our proposed estimand can be viewed as the Average Treatment Effect (ATE) conditioning on {∥ U ∥ > t } with rescaling as t increases to infinity. Detailed definition and explanation can be found in Section 3. Estimation is challenging because the limiting tail distribution is unknown and must be inferred from finite samples. To improve data efficiency and inference accuracy, we combine tail observations with moderate-frequency data in an extrapolation scheme, leveraging EVT insights alongside causal-inference techniques to achieve efficient estimation.

To the best of our knowledge, we are not aware of any work in the literature that considers this problem. In this paper, we take the first step to measure the treatment effect on extreme events. To be more specific, our contributions can be summarized as follows.

We propose a measure for the treatment effect on rare events named Normalized Extreme Treatment Effect (NETE), which essentially measures the magnitude of treatment on tailed events.

We develop two consistent estimators for NETE-a doubly robust (DR) estimator and an inverse propensity weighting (IPW) estimator-by combining recent advances in multivariate tail-dependence estimation Zhang et al. [2023] with double machine learning methodology Chernozhukov et al. [2018], and derive finite-sample, non-asymptotic error bounds.

Synthetic and semi-synthetic experiments demonstrate a good practical performance of our proposed estimator as compared to baseline estimators adapted from standard causal inference literature.

Related Work We briefly review some relevant literature in EVT and causal inference. Coles et al. [2001] provides a comprehensive introduction to EVT. A large amount of work focuses on the univariate setting Davison and Smith [1990], Leadbetter [1991], Pickands III [1975], Smith [1989]. Recently, there have been many recent works on the multivariate generalization of these results Avella-Medina et al. [2022], Zhang et al. [2023]. Causal effect estimation is a classical problem in causal inference [Rubin, 1974]. Common estimators include IPW [Rosenbaum and Rubin, 1983], DR methods [Bang and Robins, 2005, Kang and Schafer, 2007, Chernozhukov et al., 2016, 2017, 2018], Targeted Maximum Likelihood Estimation (TMLE) [van der Laan and Rubin, 2006]. There have been some efforts in the literature trying to combine the two research areas. Gissibl and Klüppelberg [2018] considers a special kind of Structural Causal Model (SCM) and shows that the proposed SCM is a kind of max-linear model. They also analyze the asymptotic distribution of their model. Chernozhukov and Du [2006], Zhang [2018], Deuber et al. [2024] consider the task of estimating the

extreme Quantile Treatment Effect (QTE). Aloui et al. [2023], Bodik [2024] assume the outcome falls into the domain of attraction and define the extreme treatment effect as the difference between EVI. Huang et al. [2024] study the estimation of extreme quantile effect and extreme average treatment effect in the continuous treatment setting. The other line of work Gnecco et al. [2021], Mhalla et al. [2020], Bodik et al. [2023] uses EVT to help causal discovery. However, we want to point out that the problems these works consider are quite different from our setting. The most similar setting would be extreme QTE estimation Chernozhukov and Du [2006], Zhang [2018], Deuber et al. [2024], but the QTE still cannot capture on how the expectation of the outcome changes under intervention. While all the previous work models extreme events using univariate variables, we model it using multivariate regularly varying variables. Moreover, our work can also be placed in the broader context in the literature that studies causal effects of treatment beyond the mean, for example, distributional effect Abadie [2002], Gautier and Hoderlein [2011], Hohberg et al. [2020], Ratio- and Log-Ratio-Based Effects Cole and Hernán [2002], VanderWeele [2013], CATE Shalit et al. [2017], Wager and Athey [2018], Abrevaya et al. [2015].

2 Preliminary

Causal Inference. We use the potential outcome framework Rubin [1974] in this paper. Let W,D,Y be the covariate, binary treatment and outcome, respectively. We denote Y ( d ) to be the potential outcome when the treatment is set to be d and assume consistency i.e., Y ( D ) = Y throughout the paper. The Average Treatment Effect (ATE) is defined as

The ATE measures the effect of a treatment on the outcome Y . In the policy-making example, D is an indicator of whether to use the policy or not. W is a covariate that may influence D , like the geographic features of a place, which will influence the local government's decision on policies, and Y can be the economic loss. The ATE in this case provides information about how much loss can be saved if a policy is enforced. Under the following exogeneity and overlap assumptions, the ATE can be identified using the g-formula E [ E [ Y | W,D = 1] -E [ Y | W,D = 0]] .

Assumption 2.1 (Exogeneity) . The data generation process satisfies ( Y (1) , Y (0)) ⊥ D | W .

Besides, the following overlap assumption is also often needed for non-asymptotic analysis.

Assumption 2.2 (Overlap) . There exists constant c p ∈ (0 , 1 / 2) such that the propensity score p ( w ) = P ( D = 1 | W = w ) ∈ [ c p , 1 -c p ] , ∀ w ∈ W .

This assumption ensures that there is no extremely high or low propensity, which can make estimators unstable.

Extreme Value Theory. The study of extremity is mainly concerned with the tail behaviors of heavy-tailed distributions, which are often modeled by the regularly varying distributions. In this paper, we modeled extremity by multivariate regularly varying distributions.

Definition 2.3. A random variable U ∈ R d + is called regularly varying with index β ∈ (0 , ∞ ) if for any norm ∥ · ∥ in R d and positive unit sphere S + = { x ∈ R d + : ∥ x ∥ = 1 } , there exists a probability measure S ( · ) on S + and a sequence b n →∞ such that n P(( ∥ U ∥ /b n , U/ ∥ U ∥ ) ∈ · ) w → c U · ν β × S for some constant c U > 0 , where · × · is the product measure and ν β ([ r, ∞ )) = r -β for all r > 0 .

The parameter γ = 1 /β is called the Extreme Value Index (EVI), which characterizes the decay rate of the tail. Notice that this definition implies that as b n → ∞ , the norm of and ∥ U ∥ and its angle U/ ∥ U ∥ become asymptotically independent. We will leverage this fact for estimation in later sections. A typical example of regularly varying distributions is the Pareto distribution.

Definition 2.4. The density of a Pareto (type II) distribution with index β ∈ (0 , ∞ ) is f ( x ) = β (1 + x ) -( β +1) , ∀ x > 0 .

Definition 2.3 implies that the rescaled norm of a regularly varying variable is asymptotically a Pareto distribution.

Notations. In the rest of the paper, we use ∥ · ∥ and ∥ · ∥ 1 as a shorthand for ℓ 1 -norm. We use the asymptotic order notation o ( · ) , O ( · ) and Θ( · ) . We use E [ · ] to represent expectation. For a matrix

A , we denote A · ,i to be its i -th column. Unif ([ a, b ]) is the uniform distribution on interval [ a, b ] and Ber ( p ) is the Bernoulli distribution with expectation p . ⊥ represents the independence relationship between two random variables.

3 Treatment Effect on Extreme Events

3.1 Extreme Semi-parametric Inference

While standard causal estimands capture average effects of D on Y , they obscure what happens in the tails-i.e., when rare, high-impact events occur. To address this, we model rare events with an explicit extreme factor U . The data we consider is of the form { ( X i , D i , Y i , U i ) } N i =1 , where D , and Y are as defined in Section 2, W = ( X,U ) is the covariate, U is the extreme part of the covariate. We use ∥ U ∥ to model the severity of rare events-large norms indicate more extreme realizations. For example, in a hurricane-loss application, U might be the vector of maximum wind speed, rainfall, and storm surge; X the region's location; D the level of infrastructure investment; and Y the resulting economic loss.

In what follows, we introduce a novel estimand that quantifies the causal effect of D on Y specifically in the tail region defined by large ∥ U ∥ . We then establish conditions for its identification under multivariate regular variation and propose two consistent estimators. We will make the following i.i.d. assumption.

Assumption 3.1. The random variables { ( X i , D i , Y i , U i ) } N i =1 are i.i.d.. Furthermore, U is regularly varying.

We are interested in the effect of treatment on the tail events of U . Similar to ATE, a naive definition of the extreme treatment effect would be

which is simply ATE conditioning on large ∥ U ∥ . However, in the case of extreme effects, the outcome may be unbounded due to the presence of extreme noise. As t increases to infinity, this effect may increase to infinity, making this quantity meaningless. Considering the climate change example, it is possible that dramatic climate change will damage or even destroy human societies, causing the effects of some policies to explode even though the policy can effectively reduce losses and slow down the process. Fortunately, regularly varying distributions have the nice property that as t increases to infinity, ∥ U ∥ /t | ∥ U ∥ > t converges weakly to the Pareto distribution (See Definition 2.3). Inspired by this property, we can normalize the quantity Y (1) -Y (0) | ∥ U ∥ > t by its growth rate. To characterize the growth of this quantity, we introduce the following polynomial growth assumption.

Assumption 3.2 (Asymptotic Homogeneous Property) . We assume that the covariate X is bounded, i.e. ∥ X ∥ ⩽ R . Let f ( X,D,U ) = E [ Y | X,D,U ] . There exists a L -Lipschitz continuous function g ( x, d, u ) and a function e ( t ) : R + → R + that satisfies lim t →∞ e ( t ) = 0 and

This assumption characterizes the growth of the outcome with respect to the extreme noise. In many real-world examples, this assumption is satisfied. For instance, research show that landslide volume often follows a power-law relationship with rainfall intensity Tuganishuri et al. [2024]; the economic loss caused by hurricanes scales polynomially with the maximum wind speed Zhai and Jiang [2014]. In these cases, f grows polynomially with respect to ∥ U ∥ and e ( t ) = 0 exactly. We define the Normalized Extreme Treatment Effect (NETE) as

where α is a known index in Assumption 3.2 from prior knowledge. Note that the previous definition (3.1) is a special case of (3.2) when α = 0 . The intuition for the scaling factor t α is that under Assumption 3.2, E [ Y ( d )] is of the order O ( ∥ U ∥ α ) and (3.2) is of the order O ( E [( ∥ U ∥ /t ) α | ∥ U ∥ > t ]) , which is finite if α < β . (3.2) implies that for a large threshold t , we have E [ Y (1) -Y (0)] ≈ t α θ NETE . Therefore, θ NETE measures the influence of treatment on the susceptibility of outcome with respect to extreme noise U .

We want to remark that NETE naturally sits at the nexus of two well-studied strands of work, tailconditional expectations in EVT, and average effects or distributional shifts at extreme quantiles, e.g., ATE and QTE. NETE can be understood as a causal analogue of EVT quantity E [ Z/t | Z > t ] , where Z is a regularly varying variable. It generalizes ATE to the setting of extreme events and aligns with the growth rate given by EVT.

3.2 Extreme Effect Identification and Estimation

The estimand (3.2) is designed to measure the treatment effect under extreme events, i.e., extremely large ∥ U ∥ . In practice, there may only be a small fraction of extreme samples in the dataset, which creates difficulties for statistical inference. To efficiently estimate the NETE, we leverage the asymptotic independence property of regularly varying variables (See Definition 2.3) to derive a novel identification formula. In particular, we have the following decomposition.

where we use Assumption 3.2 in the third equality. We can prove that the above quantity equals to

The first factor measures the average effect of treatment across different directions, while the second factor only depends on the norm of the extreme noise, which can be estimated via standard techniques in extreme value theory. We summarize the identification formula in the following proposition.

Proposition 3.3 (Identification) . Suppose that U is multivariate regularly varying and Assumption 2.1, 2.2, 3.1, 3.2 hold, we have

Proposition 3.3 separates the estimation of NETE into two parts, the expectation of the spectral measure and the index estimation, which facilitates the estimation. While in theory naive identification (3.3) works as well, we found that in practice (3.3) performs poorly (See Section 4 for empirical experiments). One reason is that without properly scaling, the (3.3) suffers from exploding ∥ U ∥ , causing larger estimation errors.

Inspired by this decomposition, we estimate the two factors separately. To make our framework more flexible, we allow an approximate scaling exponential ̂ α n as input in Algorithm 1. ̂ α n can be obtained from some prior knowledge or via other heuristics. For the first factor, we design two estimators, the Inverse Propensity Weighting (IPW) and the Doubly Robust (DR) estimators. To derive the estimators, we first randomly split the data into equal halves and use the first half for nuisance estimation, i.e., propensity and outcome. Note that the domain of the propensity function is unbounded due to the presence of U . Generally, it is difficult to estimate such a function given scarce data in the tail. To bypass this barrier, we make the following independence assumption.

Assumption 3.4. (Independence) The extreme variable U is independent of D,X , i.e., U ⊥ X,D .

Under Assumption 3.4, only X affects treatment assignment. We use the first half of data to regress ( X,D,U/ ∥ U ∥ ) on Y/ ∥ U ∥ ̂ α n to get (normalized) pseudo-outcome ̂ g and regress X on D to get an estimation of the propensity function ̂ p . Then, we use the second half for estimation. The IPW and DR estimators are defined in (3.5) and (3.6), respectively.

Notice that the second factor is the α -moment of the random variable ∥ U ∥ /t | ∥ U ∥ > t , which converges weakly to a Pareto distribution as t increases to infinity. Therefore, this quantity equals to the α moment of a standard Pareto 1 / (1 -αγ ) and the problem can be reduced to estimating the EVI of an asymptotic Pareto distribution. Here, we use the adaptive Hill estimator in (3.7) from

Algorithm 1 Algorithm for NETE Estimation

Require: Dataset D = { ( X i , D i , Y i , U i ) } n i =1 , threshold t , exponent estimation ̂ α n , estimator

1: Randomly split D into two equal parts D 1 and D 2

2: Using D 1 , estimate:

Propensity function ̂ p ( x ) via regression of D on X and clip the output of ̂ p ( x ) to the interval [ c, 1 -c ] .

Pseudo-outcome regression ̂ g ( x, d, s ) by regressing Y/ ∥ U ∥ ̂ α n on ( X,D,U/ ∥ U ∥ )

3: Define index set I = { i : ∥ U i ∥ > t, ( X i , D i , Y i , U i ) ∈ D 2 } and set S i = U i / ∥ U i ∥ for i ∈ I

4: if estimator = IPW then

5: Compute

6: else if estimator = DR then

7: Compute

8: end if

9: Compute adaptive Hill estimator on {∥ U i ∥ : i ∈ I} :

where ∥ U (1) ∥ ≥ · · · ≥ ∥ U ( k +1) ∥ and k is chosen by

10: return ̂ θ estimator n,t = ̂ η estimator n,t · ̂ µ n .

Boucheron and Thomas [2015], which provides a data-driven method for choosing the threshold. Putting the two estimations together, we get our estimator of the NETE ̂ θ · n,t = ̂ η · n,t · ̂ µ n , where the superscript · can be DR or IPW. We summarize our estimators in Algorithm 1.

Remark 3.5. Assumption 3.4 may not always hold in practice. The identification formula ( ?? ) still holds regardless of Assumption 3.4. When the assumption is violated, the propensity score based method cannot be applied here. In this case, better estimators may be a simple plug-in estimator or the DR estimator since they do not require accurate propensity estimation.

Remark 3.6. The identification formula Proposition 3.3 and our estimation methods can be easily extended for CATE estimation. We will discuss the extension in the appendix.

3.3 Non-asymptotic Analysis

Up to now, we have worked under very mild regular variation and asymptotic homogeneity conditions, which suffice to prove the consistency of our two-step estimator in the limit n, t →∞ . However, to obtain non-non-asymptotic, finite-sample deviation bounds for both the spectral-measure term and the tail-index term, we must invoke a more structured tail model. In particular, existing results such as those in Zhang et al. [2023] rely on the fact that, beyond regular variation, the noise vector behaves exactly like a (possibly linearly transformed) Pareto distribution. Although this is admittedly stronger than mere second-order regular variation, it is at present the only framework in which we can directly apply sharp concentration inequalities and Wasserstein-distance bounds for spectral-measure estimation. We therefore make the following Pareto-type assumption.

Assumption 3.7. We assume that the distribution of U comes from the following class of models

where M k = {L ( X ) : U = AZ , for A ∈ A and L ( Z ) ∈ ˜ M k } . The set of possible distributions for the components Z is

and the set of possible matrices A is

Throughout, we assume the constants satisfy m ≥ d ≥ 2 , 0 < l < 1 < u, 0 < s < 1 / 2 , σ > 0 , 0 < ξ < 1 , and ζ > 0 .

This assumption states that the extreme variable is a linear transformation of an approximate Pareto distribution. The parameter s measures how close Z is to a standard multivariate Pareto distribution. A small s means the distribution is far from Pareto.

With these assumptions, we are ready to state our main theorem, which gives a non-asymptotic rate to our estimand.

Theorem 3.8. Suppose that Assumption 2.1, 2.2, 3.1, 3.2, 3.4, 3.7 hold, α < β , where α and β are defined in Assumption 3.2 and Assumption 3.7 respectively. Furthermore, for any fixed t , with probability at least 1 -δ ,

where n t = ∑ ⌊ n/ 2 ⌋ i =1 I ( ∥ U i ∥ > t ) and R p , R g , R α are estimation errors that are monotonically decreasing with respect to sample size. Then, if n ⩾ Θ(log(1 /δ ) t β ) , with probability at least 1 -δ, δ ∈ (0 , 1 / 2) , we have

and

The error bound (3.6) consist of the nuisance error √ R p ( n/ 2 , δ ) R g ( nt -β , δ ) , variance t β/ 2 n -1 / 2 , EVI estimation error log(1 /δ ) n -1 / (2+ β ) , α error R α ( n, δ ) and bias terms t -min { 1 ,β } + t -βs/ (1 -2 s ) + e ( t ) . Similar pattern holds for (3.9). Given this general result, we choose the threshold t in a datadriven way to obtain a better rate. The idea is to use the estimated index to balance the bias and variance terms in (3.8) and (3.9). The following corollary gives the convergence rate in two different regimes.

Corollary 3.9. Under the assumptions of Theorem 3.8, further suppose that

for some c α > 0 , the following conclusions hold.

If s ∈ (0 , 1 / (2 + max { 1 , β } )) , takes t n = Θ( n (1 -2 s ) ̂ γ n ) , with probability at least 1 -δ , we have

If s ∈ [1 / (2 + max { 1 , β } ) , 1 / 2) , takes t = Θ( n ( ̂ γ n / (1+2min { 1 , ̂ γ n } ) ) , with probability at least 1 -δ , we have

Similar results hold for the IPW estimator. Due to limited space, we leave the result for IPW in the appendix. Many common machine learning algorithms, e.g., Lasso, logistic regression, neural networks, can achieve O ( n -1 / 2 ) rate in the assumption of Corollary 3.9. We want to highlight that if e ( t ) decays fast enough and become negligible compared to the other term and we know the correct scaling exponential α , Corollary 3.9 matches the rate of [Zhang et al., 2023, Theorem 3.1] without prior knowledge on the index β in the Assumption 3.7. Besides, if we have additional prior knowledge on e ( t ) and c α , we can adjust the choice of threshold t to achieve a better rate.

Remark 3.10. When the extreme noise is 1-dimensional, the spectral measure is trivially δ { 1 } and there is no need to estimate the spectral measure. Following a similar argument of Theorem 3.8 and Corollary 3.9, we can obtain a convergence rate of O ( e ( t n ) + log(1 /δ ) n -1 / (2+ β ) +log(1 /δ ) n -c α ) .

Remark 3.11. Assumption 3.7 may seem restricted at first glance. This assumption is used here because the non-asymptotic result for regularly varying extreme distributions is rare in the literature and the goal of this paper is not to develop a new estimator for the spectral measure. To the best of our knowledge, Zhang et al. [2023] is the only paper that gives such a result under Assumption 3.7. In fact, Assumption 3.7 can be easily replaced by the following two assumptions in our proof. (1) The extreme noise U is regularly varying and its norm ∥ U ∥ satisfies the von Mises condition in Boucheron and Thomas [2015]. (2) There exists an upper bound for the bias term ∥ E [ f ( U/ ∥ U ∥ ) | ∥ U ∥ > t ] -lim t →∞ E [ f ( U/ ∥ U ∥ ) | ∥ U ∥ > t ] ⩽ O ( t -c 0 ) , for some constant c 0 > 0 for a fixed Lipschitz function f . We leave this generalization to future work.

4 Experiments

Having established in Section 3 that under our regularity and overlap assumptions the DR- and IPW-based extreme treatment estimators enjoy a provable non-asymptotic error bound, we next evaluate their finite-sample behavior and compare with our estimators with naive estimators that does not consider the regularly varying structure. In what follows, Section 4.1 presents purely synthetic simulations with known NETE. Section 4.2 then moves to a semi-synthetic setting-using real noise from wavesurge datasets-to assess practical performance under realistic complexities.

4.1 Synthetic Dataset

The data generation process we use in this subsection is

where α > 0 is a constant and b ∼ N (0 , 1) , A ∼ Unif ([1 , 2] d u × d z ) . We consider two ways of generating the extreme noise. The first one follows Assumption 3.7.

We also consider a Pareto mixture, i.e., U = ( U 1 , · · · , U d u ) , U i ∼ 0 . 5 Pareto ( β ) + 0 . 5 Pareto ( β +1) . Note that Assumption 3.2 is satisfied with e ( t ) = t -α/ 2 . By Proposition 3.3, we can calculate the ground-truth effect. The graph below shows the Mean Square Error (MSE) with our estimator using different sample sizes. We take different values for α, β in the experiments. In this case, by Proposition 3.3, we know that the ground-truth NETE is 1 / (1 -α/β ) . We use Mean-Square-Error (MSE), E [( ̂ θ -θ NETE ) 2 ] , to measure the error. As a baseline, we compare our estimator with naive IPW and DR estimators. Naïve-IPW simply applies the standard IPW estimator to the U i that has norm larger than a threshold t , ignoring any tail-index modeling. Similarly, Naïve-DR augments it with the usual doubly-robust correction term but likewise ignores the Pareto structure. We leave the detailed math formulation of the baseline estimators to the appendix. The thresholds rule in Corollary 3.9 is used in the experiments and we use the same threshold selection rules for all estimators. We estimate the scaling exponential α by doing linear regression log( | Y | ) ∼ log( ∥ U ∥ ) and use the coefficient of log( ∥ U ∥ ) as ̂ α n . We leave the experiment details to the appendix. Fig. 1 and Fig. 2 show the experiment results. In the following, we use EVT-IPW and EVT-DR to represent ̂ θ IPW n,t and ̂ θ DR n,t in Algorithm 1.

Figure 1 and Fig. 2 show that under different configurations of α, β, d u , d z , our estimators generally perform better than the baseline estimators. The reason is that our estimators can make better

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use of the regularly varying structure. In general, EVT-DR achieves the smallest MSE in most experiments and is robust under different configurations. Note that the Pareto mixture does not satisfy Assumption 3.7. Fig. 2 shows that our method still maintain a good performance even if Assumption 3.7 is violated. We also observe that sometimes the MSE increases with more samples in Fig. 2. An explanation for this is that violation of Assumption 3.7 causes the threshold selection rule in Corollary 3.9 not to be applicable and the variance term dominates the error.

4.2 Semi-synthetic Dataset

Now, we use the wavesurge dataset Coles et al. [2001] to create a semi-synthetic dataset for our experiments. The wavesurge dataset has 2894 data points, which contain wave and surge heights at a single location off south-west England. Since wave and surge heights are not in the same scale and may not be positive, we shift the data and normalize each dimension by its 10% quantile. Given the wavesurget dataset, we generate our semi-synthetic dataset in the following way.

where W and S are the height of the wave and surge, respectively. In this experiment, we evaluate how well our proposed EVT-based estimators recover the Normalized Extreme Treatment Effect (NETE) when only limited 'short-term' data are available. We split the dataset into a training set (1,000 observations) and a test set (1,894 observations). First, we estimate the NETE on the training set using four estimators. Next, we apply the identification formula from Proposition 3.3 together with (4.1) to obtain a high-fidelity estimate of the NETE on the test set. Because the test-set estimate leverages additional data and the correct tail model, we treat it as a surrogate 'ground truth' for comparison. The real-world implications of this experiment is that we can use some short-term data (the training set) to predict long-term and unobserved behavior (the test set).

Table 1 shows the results we get using different estimators. The results show that our EVT-IPW and EVT-DR give estimations that are closer to the test-set estimate than the naive estimators. In particular,

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the naive estimators consistently overshoot the true NETE by an order of magnitude. In addition, while more extreme tail configurations (e.g. (1 , 3) ) slightly increase variance, the EVT-based methods remain stable, with EVT-DR deviating by at most 0.3 from the test-set estimate. These findings demonstrate that incorporating multivariate extreme value structure via our EVT-IPW and EVT-DR estimators substantially improves finite-sample estimation of treatment effects on rare, tail events, compared both to naive methods.

5 Conclusion

In this paper, we tackled the challenge of estimating treatment effects on rare, high-impact events by combining causal inference with extreme value theory. We introduced a new estimand capturing how interventions shift the tail average of outcome distributions and derived a simple identification formula using the spectral-magnitude decomposition of multivariate regular variation. Based on this, we proposed IPW and DR estimators with non-asymptotic error bounds under Pareto-type tails. Simulations and real-data experiments confirmed their advantages over naive estimators when targeting extreme outcomes. This work opens the door to more refined causal analyses in domains such as disaster risk and finance, where tail behavior matters most. A current limitation is our heuristic estimation of the tail index α ; future work will focus on theoretically grounded estimation of α and adaptive threshold selection.

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A Proofs

A.1 Identification Formula

In this subsection, we derive the identification formula in Proposition 3.3.

Proof. We first prove (3.4). By Assumption 3.2,

We next argue that lim t →∞ E [( ∥ U ∥ /t ) α | ∥ U ∥ > t ] = α/ ( β -α ) . We have

Note that ∥ U ∥ is also regularly varying. By Potter's theorem [Bingham et al., 1989, Theorem 1.56], for any ϵ > 0 and sufficiently large t , we have

Take ϵ > 0 such that α -β + ϵ -1 < -1 , we have for sufficiently large t ,

Therefore, by the dominance convergence theorem,

which implies

and (3.4) holds. We then verify the uniform integrability of function

Note that by Assumption 3.2, g is a continuous function on a compact set and thus is bounded by some constant C > 0 . We have

We have proven that the Right Hand Side (RHS) converges to a constant as t →∞ , which implies that E [ | h ( U ) | | ∥ U ∥ > t ] is uniformly bounded. We conclude that h ( U ) is uniformly integrable. By the uniform-integrability convergence theorem,

where L is the limiting distribution of ( ∥ U ∥ /t, U/ ∥ U ∥ ) | ∥ U ∥ > t and we use the asymptotic independent property of regularly varying distributions (Definition 2.3).

A.2 Non-asymptotic Analysis

To obtain a convergence rate for the estimator ̂ θ t n , we first analyze the rate of the two factors ̂ γ n and ̂ η DR n,t .

Lemma A.1. Under Assumption 3.1, 3.7 with probability at least 1 -δ , for sufficiently large n , we have

where γ = 1 /β is the EVI of U .

Proof. We adopt the non-asymptotic analysis of the adaptive Hill estimator for EVI in [Boucheron and Thomas, 2015]. In the paper, the author adopts an adaptive estimator, choosing k to be

where ̂ γ ( i ) = 1 i ∑ i j =1 log ∥ U ( j ) ∥ ∥ U ( i +1) ∥ and r n ( δ ) scales like √ log((2 /δ ) log( n )) . First, we verify the von Mises conditions in Boucheron and Thomas [2015] under Assumption 3.7. Let F be the CDF of ∥ U ∥ . By Assumption 3.7, we know that

Note that

Let ˜ Z 1 , · · · , ˜ Z m ∼ αc 1 /m (1 + z ) -α -1 , z ⩾ c 1 / ( mα ) -1 . Then, we verify the upper bound for the von Mises function, i.e., sup s ⩾ t | η ( s ) | ⩽ O ( t ρ ) for some ρ < 0 , where η is the von Mises function.

where f ( t ) is the density function of ∑ m i =1 a i ˜ z i . By [Nguyen, 2014, Theorem 2.1], we have that when ∥ U ∥ 1 > max i { a i } ζk (1 -2 s ) /α ,

Then, and

Plug in (A.3) and (A.4) into (A.2), we get

Therefore, the growth rate of the von Mises function is bounded. By [Boucheron and Thomas, 2015], with probability at least 1 -δ , we have

Lemma A.2. Undet the assumption of Theorem 3.8, with probability at least 1 -δ , we have

Proof. Let and we have

where we use the monotonicity of R p , R g .

For the second term (the bias term),

where we use Assumption 3.2 in the last equality. By the error rate assumption in Theorem 3.8,

We have the following decomposition

The first term comes from the standard statistical error of DR estimator, while the second term is the bias term caused by the finite threshold. For the first term, by standard DML theory [Foster and Syrgkanis, 2023], we have

where n t = ∑ n/ 2 i =1 I ( ∥ U i ∥ > t ) is a random variable. By Bernstein's inequality, with probability at least 1 -δ ,

Therefore, with the same probability, when n ⩾ Θ(log(1 /δ ) t β ) .

Since g is L -Lipschitz continuous, the second term is upper bounded by Wasserstein distance LW 1 ( L t U/ ∥ U ∥ , L U/ ∥ U ∥ ) , where L t U/ ∥ U ∥ is the distribution of U/ ∥ U ∥ conditioning on ∥ U ∥ > t and L U/ ∥ U ∥ is its limiting distribution as t →∞ . Therefore, we have

where we use [Zhang et al., 2023, Proposition 3.1] in the last inequality to upper bound the bias term W 1 ( L t U/ ∥ U ∥ , L U/ ∥ U ∥ ) . Combing (A.5) and (A.6), we get

Lemma A.3. Under the assumption of Theorem 3.8, with probability at least 1 -δ , we have

Proof. Similar to Lemma A.2, we have the following decomposition.

Term | η t -η | can be bounded in the same way as in the proof of Lemma A.2. By [Su et al., 2023, Theorem 1], we have

The rest of the proof is similar.

Now we are ready to prove Theorem 3.8.

Proof of Theorem 3.8. Note that by the asymptotic independence property of regularly varying distribution,

By Lemma A.1 and A.2, with high probability, ̂ γ n and | ̂ η DR n,t | is bounded. Note that by Lemma A.1,

Therefore, by Lemma A.2 and (A.7),

The bound for ̂ θ IPW n,t can be proven similarly.

Corollary A.4 (Convergence rate for IPW) . Under the assumptions of Theorem 3.8, further suppose that

for some c α > 0 , the following conclusions hold.

If s ∈ (0 , 1 / (2 + max { 1 , β } )) , takes t n = Θ( n (1 -2 s ) ̂ γ n ) , with probability at least 1 -δ , we have

If s ∈ [1 / (2 + max { 1 , β } ) , 1 / 2) , takes t = Θ( n ( ̂ γ n / (1+2min { 1 , ̂ γ n } ) ) , with probability at

Proof of Corollary 3.9. By Theorem 3.8 and the error rate assumption in Corollary 3.9, we have

If s ∈ (0 , 1 / (2 + max { 1 , β } )) , we have

Takes t = Θ( n (1 -2 s ) ̂ γ n )

Therefore, n ̂ γ n β = 1 + O ( n -1 / (2+ β ) ) . Plug this bound into (A.8) and we can get the results. Similarly, if s ∈ [1 / (2 + max { 1 , β } ) , 1 / 2) , we have

Take t n = Θ( ̂ γ n / (1 + 2 min { 1 , ̂ γ n } )) , we get the results.

B Experiment

In this section, we introduce some details of our experiments and provide additional experiments regarding the sensitivity of the algorithm with respect to the scaling parameter α .

B.1 Implementation Details

The two baseline estimators we consider are naive-IPW:

and naive-DR:

where n t = ∑ n i = ⌊ n/ 2 ⌋ +1 I ( ∥ U i ∥ ⩾ t ) and ̂ p ( X ) and ̂ g ( · ) are the estimated propensity function and the outcome function respectively. The nuisance estimation of ̂ g is obtained by running a regression Y ∼ ( X,D,U ) . We clip the propensity to [10 -4 , 1 -10 -4 ] to ensure the overlap assumption (Assumption 2.2). ϵ ∼ Unif ([ -1 , 1]) in the data generation in synthetic experiments. We use sample splitting in our experiment, using the first half for nuisance estimation. In the experiment, we use the same threshold t for all estimators, which is given by Corollary 3.9. To choose the threshold, we

t

))

.

first use the adaptive Hill estimator Boucheron and Thomas [2015] to get an estimation of EVI ̂ γ n and then set the threshold to be t = 0 . 25 n ( ̂ γ n / (1+2min { 1 , ̂ γ n } ) as in Theorem 3.8. The approximate exponential ̂ α n is coefficient of log( ∥ U ∥ ) in linear regression log( | Y | ) ∼ log( ∥ U ∥ ) . For the adaptive Hill estimator Boucheron and Thomas [2015], we follow authors' choice for hyperparameters and choose l n = 30 , r ( δ ) = √ log log( n ) and

We run logistic regression to estimate the propensity function and use random forest to model the outcome.

For the semi-synthetic experiment, we apply the same hyperparameter as above to estimate NETE. We shift the data to make it positive and normalize each dimension by its 10 % quantile. The Fig. 3 shows the rough distribution of the wavesurge data after these transformations.

Scatterplot of U

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We now describe how we calculate test-set estimation in our experiments. By the data generation process,

where we use Proposition 3.3 in the second equality. We know the ground-truth α 1 , α 2 and we can estimate the EVI γ using the test set. Suppose the estimated EVI is ̂ γ , we set the threshold to t n = 0 . 25 n ( ̂ γ/ (1+2min { 1 , ̂ γ } ) and get estimation

B.2 Additional Experiments

We conduct additional experiments under the same setup as in Section 4. Fig. 4 and Fig. 5 show the comparison of EVT-IPW and EVT-DR with and without ground-truth α . We can see that in most cases, their performance is similar, indicating that our heuristic method for estimating α is effective. Indeed, the heuristic approach even gives slightly better MSE.

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C Extension: Extreme Normalized Heterogeneous Effect Estimation

Proposition C.1 (Identification) . Suppose that U is multivariate regularly varying and Assumption 2.1, 2.2, 3.1, 3.2 hold, we have

Proof. The proof is similar to the proof of Proposition 3.3. By Assumption 3.2,

By the proof of Proposition 3.3, we know lim t →∞ E [( ∥ U ∥ /t ) α | ∥ U ∥ > t ] = α/ ( β -α ) . Thus,

Following the same argument, one can prove the RHS is uniformly bounded. By the dominance convergence theorem,

Similar to NETE estimation, we can separate the estimation of CNETE into two parts. The scaling factor E [ ∥ U ∥ α /t α | ∥ U ∥ > t ] is the same as the factor in (A.7). For the spectral part, one can use meta learners, e.g, X-learner, S-learner, T-learner, DR-learner, for estimation.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduced the Normalized Extreme Treatment Effect (NETE), and then studied its identification and estimation problems.\n\nSTRENGTHS AND WEAKNESSES:\n**Strength**\n\nThis paper is theoretically well-founded. The NETE represents a particularly interesting quantity in the context of causal inference. I have no concerns regarding the theoretical development presented in the paper.\n\n**Waekness**\n\n1. My main concern lies in the explanation of the NETE. While these quantities may represent interesting measures in causal inference, the current exposition is limited. To enhance clarity and interpretability, I suggest providing a more detailed explanation, ideally supported by visual illustrations. For instance, consider the joint distribution of (Y_1, Y_0, U) or (Y_1 - Y_0, U): under what conditions does the NETE attain large or small values? Plots or density functions illustrating these relationships would be highly informative.\n\n2. Furthermore, the explanation of the results for the semi-synthetic dataset is limited. In most parts of the semi-synthetic application, the focus is primarily on the statistical properties of the estimators. However, more detailed interpretation of the results is needed to enhance the reader’s understanding of their practical implications.\n\n\n3. From a statistical perspective, extreme value theory is well-established and well-motivated. However, researchers in the machine learning or causal inference community may be less familiar with this framework. Therefore, a more accessible explanation and stronger motivation would help broaden the impact and understanding of the work. In particular, the meaning of \\alpha may be important and warrants further clarification.\n\n4. Assumption 3.1 appears somewhat restrictive. (See question)\n\nQUESTIONS:\nAssumption 3.1 appears somewhat restrictive. It is reasonable to expect that the vector of maximum wind speed, rainfall, and storage would influence the level of infrastructure investment. The variable U could serve as a confounder. Can U be included in the covariate vector X to control for its confounding effects?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper tackles the challenging problem of estimating the impact of an intervention on rare but extreme outcomes, like extreme hurricanes or other climate events. The authors do so by combining tools from both causal inference and extreme value theory (EVT). They make several contributions:\n\n* Define key estimands when there is extreme tail risk, as well as corresponding identification and assumptions\n\n* Propose an algorithm for estimating these effects, combining both causal ML methods and EVT tools\n\n* A non-asymptotic analysis of the proposed algorithm\n\n* Several experiments and a semi-synthetic application\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n* The paper rightly highlights the need for this type of approach. The paper then does an excellent job of bridging the otherwise distinct fields of causal inference and EVT, and gives a very thoughtful discussion of the benefits and drawbacks of each approach.\n\n* The proposed algorithm and corresponding technical analysis seem to be state-of-the-art, and seem to perform well in experiments. \n\n* The paper is up front about remaining challenges and suggests many useful directions for future work. \n\n\nWeaknesses:\n* The appendix appears to be missing, which makes review more challenging. \n\n* The relationship between $Y$ and $U$ needs further clarification (see below). \n\n* Recognizing that this is challenging, there is no discussion of uncertainty quantification, which would be particularly important in a setting like this.\n\nQUESTIONS:\n* I had a hard time following the discussion in Sec 2 and 3.1 that defines the key quantities of interest, and especially the relationship between U and the other variables. The most intuitive explanation is the sentence on line 124 “for example …”. But it would be useful to give more clarity on the relationships between U and Y especially. e.g., do you think of Y as having a structural component and U noise? The synthetic dataset in Sec 4.1 is a useful step. \n\n* For the semi-synthetic dataset, does $Y$ correspond to anything concrete? Is there any notion of uncertainty that can help calibrate whether these point estimates are “large” or “small”?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-20"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose a principled approach to estimate causal effects conditional on threshold exceedances in a heavy-tailed data-generating process. Their method is based on a model in which data is generated as a function of a Pareto-type intensity random variable. The causal effect is quantified as the rescaled difference in means between treatment groups. The authors introduce multiple estimators and analyze their non-asymptotic theoretical performance. These results are illustrated through extensive experiments comparing the performance of the proposed estimators for the so-called NETE.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\nThe methodology is clear, the results are sound and mathematically well-founded. The contribution is both novel and timely.\n\n### Weaknesses\n\n- Existence of the mean: One key reason why existing approaches for estimating causal effects in the extremes avoid analyzing the Expected Treatment Effect (ETE) is that this quantity does not always exist-unlike quantiles, which are commonly used in such settings. This is why the authors focus on the NETE instead of the ETE. However, beyond the mathematical convenience, it remains unclear what kind of relationships the NETE can capture. In multivariate regular variation, similar standardization techniques are used to quantify dependence. No analogous interpretation is provided in the present work.\n- Relevance of the synthetic data experiment: It is not clear from the experiment what specific insights the NETE provides that could not be obtained through a classical causal analysis-assuming the mean exists. The added value of NETE over traditional estimands should be better illustrated or clarified. In partuclar, the claim on l.120-121 is unclear: in general when studying a mean effect, the average, if it exists, it highly impacted by rare events (and not obscured). This point would benefit from further explanations.\n\nQUESTIONS:\n- Would it be possible to provide deeper insight into what the NETE represents and what type of causal relationship it captures?\n- Similarly, could a synthetic experiment be constructed where the NETE identifies a causal relationship that would not be detected by a classical causal inference method (again, assuming the mean exists)?\n- Where is the proof for Proposition 3.3? Not sure see why this holds.\n\nMore generally:\n- Line 13+: The term “extremities” is somewhat awkward, as it can imply an upper bound. Consider using “extremes” or “extreme events” instead.\n- Line 84: The term “consistency” is unclear in this context. Please clarify whether it refers to consistency of an estimator, or something else.\n- Lines 94 and 104: Is this the same constant c? If so, it should be explicitly stated; if not, the notation should be differentiated.\n- Lines 91 and 115: The mean is used before it is formally defined. Consider introducing the necessary definitions earlier for clarity.\n- Line 91: The orthogonality symbol is used without explanation. Please define this notation when it first appears.\n- Line 94: The term “estimated propensity” is vague—does it refer to an estimated propensity score? If so, please make this explicit.\n\nLIMITATIONS:\nLimitations were detailed in the weaknesses section.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a thorough theoretical analysis of estimating average treatment effects under extreme noise conditions. It adapts tail estimators to the causal inference setting, generalizes identifiability theorems under commonly assumed conditions, and supports its results with experiments on synthetic (and semi-synthetic) data.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n\n- I appreciate the motivation to estimate treatment effects under extreme scenarios which makes the scenario studied by the paper relevant and unique to the best of my knowledge.\n- The theoretical analysis is thorough. \n\n### Weaknesses\n\n- Since there is no widely agreed-upon definition of extreme treatment effects in the literature, and the authors appear to propose a new one, it would be helpful to state this more clearly in the abstract (e.g., \"introducing a new definition to model extremities in treatment effects\"). \n\n- The sentence *\"Classic causal inference literature mainly focuses on estimating the average effects among certain groups\"* gives the impression that no prior work has modeled quantities beyond the population average (ATE). However, there is a substantial body of work on estimating conditional average treatment effects (CATE) [1,2], modeling distributions [3,4], and on tail/extreme treatment effects [5,6,7].\n\n- As previously mentioned, a quick search reveals at least two papers available online that tackle the intersection of Extreme Value Theory (EVT) and causal inference in similar (though not identical) settings — namely, Huang et al. (2022) and Aloui et al. (2023). A more thorough literature review focusing on this intersection is necessary. Both papers cite additional related works. While the current paper focuses on the average treatment effect under heavy-tailed exogenous variables — a different setting — the use of EVT theory is highly relevant and similar. A discussion of these differences would strengthen the literature review.\n\n- The authors only study an average (aggregate) quantity, not one conditioned on covariates. I believe conditioning on covariates would make the contribution more complete and relevant. For example, in healthcare, we may care about the extreme response of a treatment for a specific individual, such as  \n\n $$\\theta^{\\text{NETE}}(X) = \\lim_{t \\rightarrow \\infty} \\mathbb{E}\\left[\\frac{Y(1) - Y(0)}{t^{\\alpha}} \\middle|\\ X, |U| > t\\right]$$\n\nDo the authors see a natural extension of their work to this conditional setting?\n\n- In Assumption 3.2, I find the statement is unclear. Should it state that there exists some $\\alpha$ ? Is $\\alpha$ unique for a fixed $g$? Also, is $e(t)$ a function of $g$? That is, should the statement read: `` $\\exists g, \\exists e$ such that …”? Clarification is needed for me to understand the statement better, and $\\alpha$ should be more explicitly defined. I recall seeing similar assumptions in EVT literature — could the authors point to any related work? Also, does the property hold only as $t \\to \\infty$, or for all $t$?\n\n- Regarding the statement *“E[Y(d)] is of the order $O(\\|U\\|^{\\alpha})$ and (3.2)”* — does this mean that $\\alpha$ is fixed for each $Y$? Please clarify.\n\n- The sentence *“In these cases, $f$ grows polynomially with respect to $\\|U\\|$ and $e(t) = 0$ exactly.”* refers to just one choice of \\(e\\). However, many functions \\(e(t)\\) could verify the required property, even if they are nonzero but converge to zero as $t \\to \\infty$. Is my understanding correct?\n\n- The claim that the definition *“generalizes ATE to the setting of extreme events and aligns with the growth rate given by EVT”* is somewhat misleading. It is not obvious how this is a generalization of ATE. Rather, it appears to be a separate definition. It may be more accurate to say it generalizes the *unnormalized* version (ETE), if anything. Or am I missing something? \n\n- Could the authors add error bars in Figure 2? The experiments are repeated over 50 different seeds, meaning the data changes each time, and reporting variance would be helpful.\n\n- The authors mention that as the number of samples increases, the assumption becomes violated and the estimator becomes dominated by error. This is quite counterintuitive: if the estimator is consistent, it should converge to the true estimand as the sample size grows. Could the authors elaborate on this phenomenon? Is there a way to filter out “bad samples” that lead to this behavior?\n\n- A final question: Could the authors expand on how they envision extending this work to the estimation of conditional average treatment effects? Also, what do they expect regarding the performance of their estimator when conditional unconfoundedness is violated? While this question likely deserves a dedicated study, if the authors have time during the rebuttal phase, it would be informative to include a synthetic experiment where $X \\sim \\text{Unif}[0,1]^d$ with $d$ ranging from 5 to 10, and only the first 5 dimensions used to adjust for bias (as is done now). How would the performance degrade as dimensionality increases (i.e., as more confounders are missing)? While not a formally defined sensitivity analysis, this could still help us understand the robustness of the method to unobserved confounding.\n\n\n#### References\n\n[1] Shalit, Uri, Fredrik D. Johansson, and David Sontag. \"Estimating individual treatment effect: generalization bounds and algorithms.\" International conference on machine learning. PMLR, 2017.\n\n[2] Abrevaya, Jason, Yu-Chin Hsu, and Robert P. Lieli. \"Estimating conditional average treatment effects.\" Journal of Business & Economic Statistics 33.4 (2015): 485-505.\n\n[3] Gautier, Eric, and S. T. E. F. A. N. Hoderlein. \"Estimating the distribution of treatment effects.\" arXiv preprint arXiv:1109.0362 (2013).\n\n[4] Hohberg, Maike, Peter Pütz, and Thomas Kneib. \"Treatment effects beyond the mean using distributional regression: Methods and guidance.\" PloS one 15.2 (2020): e0226514.\n\n[5] Huang, Wei, Shuo Li, and Liuhua Peng. \"Extreme continuous treatment effects: Measures, estimation and inference.\" arXiv preprint arXiv:2209.00246 (2022).\n\n[6] Aloui, Ahmed, et al. \"Treatment Effects in Extreme Regimes.\" arXiv preprint arXiv:2306.11697 (2023).\n\n[7] Bodik, Juraj. \"Extreme treatment effect: Extrapolating dose-response function into extreme treatment domain.\" Mathematics 12.10 (2024): 1556.\n\nQUESTIONS:\nI added the questions in the weaknesses above, please read above for further details. \nThree things that, in my opinion, would significantly increase the value of the paper:\n\n- A more thorough literature review  \n- Clearer mathematical statements (it was sometimes difficult to identify the definition of each symbol)  \n- Experiments: discussion of the effect of missing confounders on stability, explanation of stability with respect to the number of samples, and inclusion of error bars (especially as a function of the number of samples, to test the hypothesis mentioned by the authors)\n\nLIMITATIONS:\nYes. I don't think any important limitation is missing as the authors state the assumptions under which their analysis hold.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe work introduces Normalized Extreme Treatment Effect (NETE), a causal estimand that targets how an intervention shifts tail outcomes—e.g. damage during 100‑year storms—rather than means. Building on multivariate regular variation, the authors:\n\n- formalise NETE and prove identifiability under potential‑outcome exogeneity plus an “asymptotic homogeneity” growth condition\n\n- derive a decomposition that separates spectral‑measure and α‑moment terms, enabling plug‑in estimation\n\n- craft two estimators—IPW and doubly‑robust (DR)—combined with an adaptive Hill tail‑index estimator \n\n- provide non‑asymptotic error bounds under a structured Pareto‑mixture model \n\n- validate the approach on extensive synthetic grids and a wave‑surge semi‑synthetic dataset, where EVT‑DR consistently attains the lowest MSE versus naïve baselines\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n- Novel Problem Framing. NETE fills a genuine gap by marrying EVT with causal inference to quantify tail counterfactuals—distinct from existing QTE work.\n\n- Methodological Rigor. Doubly‑robust construction and adaptive Hill estimation yield consistency plus finite‑sample rates.\n\n- Synthetic Thoroughness. Four‑way sweeps over tail index, dimension, and growth exponent highlight estimator robustness.\n\n- Clear Mathematical Exposition. Assumptions and steps are easy to trace; proofs provided.\n\n### Weaknesses & Questions for the Authors\n1. Observability of Extreme Noise U. Real‑world practitioners seldom measure the latent driving shock; instead Y alone reflects severity. How realistic is it to assume ∥U∥ is observed so that {∥U∥ > t} can be conditioned on? Could the method be adapted when only Y (or a proxy) identifies extremes?\n\n2. Independence Assumption. Assumption 2.1 treats U independent of treatment assignment after conditioning on X. In many policy settings the exposure and hazard covary (e.g. flood defences built where storms frequent). Please discuss robustness / sensitivity when this fails.\n\n3. Tail‑Model Specification. Non‑asymptotic bound (Thm 3.5) hinges on Pareto‑mixture Assumption 3.4; yet Fig. 2 shows violation degrades MSE. Can the authors provide a diagnostic or adaptive procedure to detect mis‑specification in practice?\n\n4. Scaling‑Exponent α. α is estimated by a log‑log regression; no theory is provided for its error propagation. Empirically, how sensitive is NETE to mis‑estimated α?\n\n5. Compute & Reproducibility. Please provide (i) code, (ii) random seeds, (iii) hardware spec, and (optionally) (iv) wall‑clock times so reviewers can replicate Tables 1–2.\n\n6. Real‑world Case Study. A genuine observational dataset—e.g., hurricane damage with/without building codes—would greatly strengthen practical relevance.\n\n7. Why are only averages shown in Fig 2 although the experiment was repeated 50 times. Please add CI.\n\n8. A paragraph on Limitations is missing.\n\nQUESTIONS:\nPlease see the section on strengths and weaknesses.\n\nLIMITATIONS:\nLimitations have not been discussed.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}