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A Location-Invariant Estimator of Extremal Quantile Treatment Effects for Heavy-Tailed Distributions

Xin Yu, Shuwei Huang, Jicheng Liu, Jielin Tang, Bolin Wang, Yunxiao Zhang, Tian Zhao

arXiv:2609.04018v2cs.LGstat.APstat.ME

TL;DR

Extreme QTEs can target quantiles beyond the observed range, but existing extrapolation-based estimates are not invariant to a common location shift. The paper combines a causal, location-invariant Fraga EVI estimator with difference extrapolation; simulations confirm location invariance, threshold stability, and confidence-interval coverage.

  • Problem

    Existing extremal QTE estimators for heavy-tailed outcomes are not location invariant, although the population QTE is, and extreme quantiles may lie beyond the observed data.

    Source: §1 · p. 2

  • Method

    The paper adapts the location-invariant Fraga EVI estimator through inverse propensity score weighting and combines it with difference extrapolation, which cancels location shifts in quantile differences.

    Source: §1 · p. 3

  • Results

    Simulations verify the proposed estimator’s location invariance, stability across threshold k, and coverage of its confidence intervals.

    Source: §6 · p. 17

  • Takeaways & Limitations

    The consistent analytical variance estimator yields asymptotically honest confidence intervals for the proposed extremal QTE estimator.

    Source: §6 · pp. 16–17

    The Fraga-type estimator has relatively large variance, and the auxiliary level βn lacks a formal optimality theory for its choice.

    Source: §6 · p. 17

Abstract

from arXiv · show

Quantile treatment effects (QTEs) measure the effect of a treatment on the distribution of an outcome, and their estimation at extreme quantile levels is of central interest in applications where the target quantiles lie far beyond the range of the data. For heavy-tailed potential outcomes, existing extremal QTE estimators rely on extrapolation combined with a causal extreme value index (EVI) estimator, but the resulting estimator is not invariant under a common location shift of the potential outcome distributions, even though the population QTE is. We address this issue in two steps. First, we adapt the location-invariant Fraga estimator of the EVI to the causal setting using inverse propensity score weighting. Second, we replace the original extrapolation formula with a difference-based scheme, under which the location parameter cancels when quantile differences are taken. The resulting QTE estimator is therefore location invariant. We establish the consistency and asymptotic normality of the proposed extremal QTE estimators, and provide a consistent variance estimator, leading to asymptotically valid inference. A simulation study confirms the location invariance, the stability with respect to the threshold, and the coverage of the proposed methods.

1 Introduction

QTEs capture how treatment effects vary across an outcome distribution, and extreme quantiles matter when outcomes of interest lie beyond observed data. The paper addresses location dependence in existing extremal QTE estimation with a location-invariant EVI estimator and difference-based extrapolation.

1 Introduction

  • Existing extremal QTE estimation targets heavy-tailed outcomes beyond the observed range, where fixed- and intermediate-level QTE estimators are not applicable.The extreme-level regime has nτn → a ≥ 0.Source: §1 · p. 2
  • The estimator of Deuber et al. is not location invariant because both its causal Hill EVI estimator and its multiplicative extrapolation formula depend on location.The population QTE is unchanged by a common location shift of both potential outcome distributions.Source: §1 · p. 3
  • The causal Fraga estimator adapts Fraga Alves’s location-invariant EVI estimator through inverse propensity score weighting and has consistency and asymptotic normality under weaker conditions than the causal Hill estimator.The paper combines it with difference extrapolation, where the location parameter cancels in quantile differences.Source: §1 · pp. 3–4
  • The proposed difference extrapolation combined with the causal Fraga estimator remains stable across location shifts, unlike the estimates of Deuber et al..Figure 1 illustrates this comparison for model H2 at n = 5000 and τn = 5/(n log n).Source: Figure 1 · p. 2, §1 · p. 3
  • The paper establishes asymptotic normality for its causal-Fraga-based QTE estimators, provides a consistent variance estimator, and reports simulations confirming threshold stability and confidence-interval coverage.The introduction also situates the method within causal research on extreme events and describes its sections on theory, estimation, and simulation.Source: §1 · pp. 3–4

2 Preliminaries

The preliminaries set out the heavy-tail framework for extreme-value extrapolation and the potential-outcome conditions used to identify and estimate QTEs. They also distinguish the fixed, intermediate, and extreme quantile regimes.

2.1 Extreme Value Theory

  • Extreme quantiles are studied at levels 1 − τn with τn → 0 and nτn → a ≥ 0; heavy-tailed outcomes have EVI γ > 0, which governs tail heaviness.The extreme-value framework uses maximum-domain-of-attraction and regular-variation conditions; second-order regular variation supplies an additional rate condition for asymptotic normality.Source: §2.1 · pp. 4–6
  • The EVI is unchanged by location shifts, while the Fraga and Pickands estimators are location invariant because they use order-statistic differences or shifts.The Hill and moment estimators are not location invariant; the paper focuses on the heavy-tailed regime γ > 0.Source: §2.1 · pp. 4–5

2.2 Quantile Treatment Effect

  • The QTE is defined from the potential-outcome distributions, and observational identification relies on unconfoundedness and common support.Inverse propensity score weighting is used in the fixed-quantile estimator described in the preliminaries.Source: §2.2 · pp. 6–7
  • At intermediate levels, the estimated QTE is asymptotically normal, whereas at extreme levels with nτn → a ≥ 0 that normality fails without tail extrapolation.The dynamic-QTE framework requires additional potential-outcome tail assumptions, including membership in an extreme-value domain of attraction.Source: §2.2 · p. 7

3 Causal Extreme Value Index Estimator

This section introduces a causal, location-invariant Fraga estimator of the extreme value index (EVI), then establishes its consistency and asymptotic properties. The estimator adapts Fraga Alves’s construction using inverse propensity score weighting.

3 Causal Extreme Value Index Estimator

  • The causal Fraga estimator adapts Fraga Alves’s location-invariant EVI estimator with inverse propensity score weighting to address confounding.Source: §3 · p. 8

3.1 Definition of the Causal Fraga Estimator

  • For each treatment arm, it estimates γ_j from weighted tail observations using quantiles at primary α_n and auxiliary β_n thresholds.Source: §3.1 · p. 8
  • A common additive shift leaves the estimator unchanged; unlike causal Hill, it is not sensitive to threshold location.Source: §3.1 · p. 8

3.2 Asymptotic Properties of the Causal Fraga Estimator

  • Consistency holds under more relaxed conditions than for causal Hill, while asymptotic normality requires second-order regular variation and additional threshold-rate conditions.Source: §3.2 · pp. 8–9
  • The joint asymptotic distribution of the treated- and control-arm estimators supplies input for extremal-QTE analysis.Source: §3.2 · pp. 9–10

4 Extremal Quantile Treatment Effect Estimator

For extreme quantile levels, the paper replaces multiplicative extrapolation with a difference-based estimator built on the causal Fraga EVI estimator. It establishes asymptotic inference for this location-invariant estimator, subject to tail and threshold-rate conditions.

4.1 Definition of the Extremal QTE Estimator

  • At extreme levels, empirical quantiles are severely biased because only finitely many expected observations exceed the target level.Source: §4.1 · p. 10
  • Difference extrapolation cancels the location parameter and, combined with a location-invariant causal Fraga estimator, yields a location-invariant extremal QTE.Source: §4.1 · pp. 10–11

4.2 Asymptotic Properties of the Extremal QTE Estimator

  • The extreme-level regime permits nτ_n→0, but τ_n cannot decay arbitrarily fast relative to β_n without destabilizing extrapolation.Source: §4.2 · p. 11
  • Under conditions including comparable potential-outcome tails, the extremal QTE estimator is asymptotically normal and its variance estimator is consistent.Source: §4.2 · pp. 11–12
  • The difference construction also works with other causal EVI estimators meeting the required convergence rate, but full location invariance requires the EVI estimator itself to be location-invariant.Source: §4.2 · p. 12
  • The difference-extrapolation estimator is location-invariant and first-order unbiased, with a consistent variance estimator and asymptotically valid confidence intervals.Source: §4.2 · p. 12

5 Simulations

The simulations assess location-shift invariance, threshold sensitivity, and confidence-interval coverage across heavy-tailed models. Only the causal Fraga estimator with difference extrapolation produces QTE estimates stable under growing location shifts, and it behaves stably across thresholds.

5.1 Experimental Setup

  • The simulations compare causal Hill and causal Fraga estimators with multiplicative and difference extrapolation across heavy-tailed models, location shifts, thresholds, and extreme quantile levels.The study examines location-shift effects, sensitivity to k, and confidence-interval coverage.Source: §5.1 · p. 13

5.2 Simulation Results

  • Across H1–H3, only causal Fraga with difference extrapolation keeps QTE estimates stable as the common location shift u grows.The distributions of estimates are shown for H1, H2, and H3.Source: §5.2 · p. 14, Figure 2 · p. 14, Figure 3 · p. 15
  • Difference extrapolation reduces variance inherited from the EVI estimator but adds variance from estimating the intermediate quantile at βn.Causal Fraga estimates are generally more variable, reflecting the larger variance of its EVI estimator.Source: §5.2 · p. 14
  • Across models, causal Fraga with difference extrapolation behaves stably across k, while causal Hill variants fluctuate under H1 and H2 and spike as k increases.The Hill-based spikes reach extremely negative values, truncated by the logarithmic scale.Source: §5.2 · p. 14, Figure 5 · p. 16, Figure 6 · p. 16
  • Bootstrap intervals for causal Fraga with difference extrapolation have coverage close to nominal, while its analytical interval is closer to nominal at n = 5000 than at n = 1000.For causal Fraga with multiplicative extrapolation, inflated bootstrap standard errors at n = 5000 yield coverage above nominal.Source: §5.2 · p. 15
  • For H2 at n = 5000 and τn = 5/(n log n), Deuber et al. bootstrap coverage falls rapidly as u grows, while the proposed estimator’s coverage remains stable.The comparison concerns the empirical coverage of bootstrap confidence intervals.Source: §5.2 · pp. 15–16

6 Conclusion

The conclusion presents a location-invariant extremal QTE estimator and reports its asymptotic properties and simulation findings. Appendix A.1 recalls the triangular-array results used in the proofs.

6 Conclusion

  • The causal Fraga estimator and difference extrapolation scheme yield a location-invariant extremal QTE estimator under unconfoundedness.The conclusion also reports consistency and asymptotic normality for the estimators and consistency of an analytical variance estimator.Source: §6 · p. 16
  • Simulations verified location invariance, stability across threshold k, and coverage of the associated confidence intervals.The figures describe threshold-dependent mean estimates for H3, coverage for H1–H3, and coverage under location shifts for H2.Source: §6 · p. 17, Figure 7 · p. 17, Figure 8 · p. 17
  • Future work could improve precision using lower-variance location-invariant EVI estimators and develop a formal optimality theory for choosing auxiliary level β_n.The Fraga-type estimator has relatively large variance by construction, and β_n currently follows an adaptive rule.Source: §6 · p. 17

A.1 Asymptotic Theory

  • The appendix recalls the triangular-array weak law and Lindeberg central limit theorem, plus the continuous mapping theorem, for the subsequent proofs.Source: §A.1 · p. 21

A.2 Proof of Theorem 1

This section develops proofs for the extremal quantile estimator, establishing consistency for both treatment arms and advancing the asymptotic normality arguments. The proofs use tail-quantile approximations and estimated-quantile control, then invoke a Lindeberg CLT and Slutsky’s theorem under additional assumptions.

A.2 Proof of Theorem 1

  • The proof establishes relative consistency of the estimated treatment-arm quantile at level 1−αn, with its relative error δn=op(1).This control is used in the consistency argument for the estimator.Source: §A.2 · pp. 27–28
  • Under Assumptions 1–3, the estimator converges in probability to γ1 for treatment arm j=1, with the control-arm proof following by symmetric substitution.The proof sandwiches its dominant term around γ1 and shows the remaining terms vanish.Source: §A.2 · pp. 27, 29

A.3 Proof of Lemma 1

  • For the CLT proof of Lemma 1, A1 is identified as the dominant term requiring refined analysis, and its log-ratio is expanded using second-order regular variation.The expansion’s op(1) remainder is uniform across qualifying summands because the regular-variation approximation is uniform over the relevant sliding window.Source: §A.3 · pp. 29–30
  • The proof establishes the Lindeberg condition and applies the Lindeberg CLT.Source: §A.3 · p. 31
  • The proof identifies b1 = γ1/(1 + γ1) and c1 = 1/(1 − ρ1) as bias constants.Source: §A.3 · p. 32

A.4 Proof of Theorem 2

  • Under Theorem 2’s additional assumptions, the proof treats the two bias terms and remainder, then invokes Slutsky’s theorem.Source: §A.4 · p. 32

A.5 Proof of Theorem 3

The proof establishes joint asymptotic normality by showing cross-covariance vanishes and verifying a Lindeberg condition for the Cramér–Wold argument.

A.5 Proof of Theorem 3

  • Cross-covariance vanishes because treatment indicators for the two groups cannot both be nonzero for the same observation.The proof states that the resulting cross-covariance tends to zero.Source: §A.5 · p. 33
  • The Cramér–Wold step uses independent mean-zero summands and verifies Lindeberg after controlling the standardized summands.Source: §A.5 · p. 34
  • The Cramér–Wold theorem yields joint convergence, which is the claimed joint asymptotic normality.Source: §A.5 · p. 34

A.6 Proof of Lemma 2

The proof analyzes the difference extrapolation formula through three terms, using intermediate-level ratios, regular variation, and termwise asymptotic bounds.

A.6 Proof of Lemma 2

  • The proof defines d_n = α_n/τ_n, r_n = α_n/β_n, and e_n = β_n/τ_n, with log e_n/√k0 → 0.Source: §A.6 · p. 34
  • The quantile-estimation error term is shown to be o_p(1) after scaling, using log e_n → ∞ and k0/k → 0.Source: §A.6 · p. 35
  • The proof treats the core term by an integral representation and controls the second-order bias using regular variation before combining the three terms.Source: §A.6 · pp. 35–36

A.7 Proof of Theorem 4

The proof derives the asymptotic distribution of the extremal QTE by combining the two treatment-arm estimation errors. The arm-specific terms are asymptotically independent, yielding the stated variance.

A.7 Proof of Theorem 4

  • The treatment-arm terms are jointly asymptotically normal with a diagonal covariance matrix, so they are asymptotically independent.The covariance matrix has diagonal entries γ1^2a1^2 and γ0^2a0^2.Source: §A.7 · p. 36
  • The QTE error expansion combines the estimated quantile errors from both treatment arms.The proof applies Lemma 2 to each arm.Source: §A.7 · p. 36
  • The arm-specific terms S1 and S0 enter with weights q1 max{q1, q0} and q0 max{q1, q0}, converging respectively to min{1, κ} and min{1, 1/κ}.Source: §A.7 · p. 37

A.8 Proof of Theorem 5

The proof establishes the asymptotic behavior of the variance estimator by reducing its empirical components to oracle quantities. Estimated propensity scores and threshold indicators introduce uniformly negligible errors under the stated conditions.

A.8 Proof of Theorem 5

  • The estimated per-observation quantity uses the estimated propensity score and causal Fraga EVI estimator, and is compared with its oracle counterpart.The proof establishes that their difference is uniformly op(1) on the tail event.Source: §A.8 · pp. 37–38
  • The log-quantile expansion has a uniformly negligible remainder under Theorem 2’s bias-decay conditions.The proof combines this expansion with the Fraga EVI estimator to control the empirical remainder.Source: §A.8 · p. 38
  • Propensity-score replacement and tail-indicator replacement contribute op(1) error, with the βn terms canceling in the subtraction.The proof controls the propensity-score replacement under Assumption 1 and the indicator replacement using threshold consistency.Source: §A.8 · p. 38
  • The proof bounds the average absolute oracle remainder by Op(βn), supporting the variance-estimator consistency argument.The empirical and oracle remainders are then bounded together.Source: §A.8 · p. 38

B Multiplicative Extrapolation

The appendix derives asymptotic inference for the conventional multiplicative extrapolation estimator, which is not location-invariant. In simulations, difference extrapolation is advantageous with the causal Fraga estimator when its variance is large, while the causal Hill estimator shows a slight disadvantage.

B Multiplicative Extrapolation

  • The cited passage states that, by Theorem 3, the pair (S1, S0) is jointly asymptotically normal, while the supplied excerpt does not give the complete covariance matrix.Source: B Multiplicative Extrapolation · p. 41
  • The resulting asymptotic normality and consistent variance estimator yield an asymptotic confidence interval for the multiplicative estimator.This estimator is not location-invariant and serves as the baseline for comparison with difference extrapolation.Source: B Multiplicative Extrapolation · pp. 38, 42

C.1 Squared Errors of QTE estimations under the Location Shift

  • With a high-variance causal Fraga EVI estimator, difference extrapolation is advantageous; with a low-variance causal Hill estimator, it is slightly inferior to multiplicative extrapolation.The Hill comparison reflects the additional variance from estimating the intermediate quantile at βn.Source: §C.1 · p. 42

C.2 Mean Squared Errors with respect to k

Across the three models and the full range of k, difference extrapolation with the causal Fraga estimator has low, stable mean squared error. The causal Hill variants are erratic under H1 and H2, while multiplicative extrapolation with causal Fraga has large errors under H2 and H3.

C.2 Mean Squared Errors with respect to k

  • Difference extrapolation with the causal Fraga estimator attains low, stable mean squared error across H1–H3 and the whole range of k.The figures show mean squared error as a function of threshold parameter k for each model.Source: §C.2 · p. 42, Figure 15 · p. 44, Figure 16 · p. 44
  • Both causal Hill extrapolation variants have large, erratic mean squared errors under H1 and H2, consistent with Hill EVI instability in those settings.The discussion relates this instability to Fraga Alves.Source: §C.2 · p. 42
  • Under multiplicative extrapolation, the causal Fraga estimator’s high variance, amplified at small k, inflates variance and bias components under H2 and H3.The figures plot mean squared error against k for these models.Source: §C.2 · p. 42, Figure 16 · p. 44, Figure 17 · p. 45
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