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Tail-Calibrated Estimation of Extreme Quantile Treatment Effects

Mengran Li, Daniela Castro-Camilo

Abstract

Extreme quantile treatment effects (eQTEs) measure the causal impact of a treatment on the tails of an outcome distribution and are central for studying rare, high-impact events. Standard QTE methods often fail in extreme regimes due to data sparsity, while existing eQTE methods rely on restrictive tail assumptions or on interior-quantile theory. We propose the Tail-Calibrated Inverse Estimating Equation (TIEE) framework, which combines information across quantile levels and anchors the tail using extreme value models within a unified estimating equation approach. We establish asymptotic properties of the resulting estimator and evaluate its performance through simulation under different tail behaviours and model misspecifications. An application to extreme precipitation in the Austrian Alps illustrates how TIEE enables observational causal attribution for very rare events under anthropogenic warming. More broadly, the proposed framework establishes a new foundation for causal inference on rare, high-impact outcomes, with relevance across environmental risk, economics, and public health.

Tail-Calibrated Estimation of Extreme Quantile Treatment Effects

Abstract

Extreme quantile treatment effects (eQTEs) measure the causal impact of a treatment on the tails of an outcome distribution and are central for studying rare, high-impact events. Standard QTE methods often fail in extreme regimes due to data sparsity, while existing eQTE methods rely on restrictive tail assumptions or on interior-quantile theory. We propose the Tail-Calibrated Inverse Estimating Equation (TIEE) framework, which combines information across quantile levels and anchors the tail using extreme value models within a unified estimating equation approach. We establish asymptotic properties of the resulting estimator and evaluate its performance through simulation under different tail behaviours and model misspecifications. An application to extreme precipitation in the Austrian Alps illustrates how TIEE enables observational causal attribution for very rare events under anthropogenic warming. More broadly, the proposed framework establishes a new foundation for causal inference on rare, high-impact outcomes, with relevance across environmental risk, economics, and public health.
Paper Structure (36 sections, 4 theorems, 62 equations, 3 figures, 4 tables)

This paper contains 36 sections, 4 theorems, 62 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Existing approaches to quantile treatment effect estimation
  4. Extreme quantile treatment effects
  5. The inverse estimating equation framework
  6. Tail-calibrated inverse estimating equation (TIEE) framework
  7. From inverse estimating equations to tail calibration
  8. Causal signal functions
  9. Tail modelling and extrapolation
  10. Numerical solution for the TIEE estimator $\hat{\theta}_{d,\tau}$
  11. Theoretical properties
  12. Identification, uniqueness, consistency and asymptotic normality
  13. Inference for the extreme quantile treatment effect $\delta(\tau)$
  14. Simulation Study
  15. Data-Generating Process
  16. ...and 21 more sections

Key Result

Theorem 1

Let $Y$ be a real-valued random variable with continuous marginal distribution function. Suppose the target quantile $\theta_{d,\tau}$ is uniquely defined by $F_{Y_d}(\theta_{d,\tau})=\tau$ for $\tau\in(0,1)$. Assume that for every $(W,\tau,\zeta)$ the function $g_{\mathrm{TIEE},d}(W,\theta,\tau,\ze

Figures (3)

  • Figure 1: Coverage rates of 90% confidence intervals for the extreme quantile treatment effect. The comparison includes Zhang, Hill, and TIEE estimators with varying sample fractions ($5/n$, $1/n$, and $5/(n \log n)$). The dashed horizontal line represents the nominal confidence level. The top row shows results for the heavy-tailed scenarios, while the bottom row shows results for the light-tailed scenarios.
  • Figure 2: Mean squared error of the TIEE-IPW estimator of the EQTE $\delta(\tau_n)$ as a function of the intermediate quantile level $p_u$ for $(1-\tau_n)\in\{5/\log(n), 1/n, 5/n\}$.
  • Figure 3: Mean squared error (MSE) of the TIEE-IPW estimator of the EQTE $\delta(\tau_n)$ as a function of the grid size $K$ for $(1-\tau_n)=5/\log(n), 1/n, 5/n$.

Theorems & Definitions (6)

  • Theorem 1: Identification and Uniqueness
  • proof
  • Lemma 1: Glivenko--Cantelli property of the integrated signal
  • Theorem 2: Consistency
  • proof
  • Theorem 3: Asymptotic Normality