Rerandomization for QTE Estimation

• The paper presents a novel integration of rerandomization into QTE estimation by enforcing covariate balance via the Mahalanobis distance.
• It reveals that rerandomization induces a non-Gaussian limiting distribution while yielding efficiency gains, with variance reductions estimated up to 45%.
• Monte Carlo simulations and empirical studies validate shorter confidence intervals and robust coverage, highlighting practical improvements in causal inference.

Rerandomization for quantile treatment effect (QTE) estimation concerns the integration of rerandomization procedures—designed to enforce covariate balance during experimental treatment assignment—into the estimation and inference of quantile-based causal effects. While complete randomization is a classical standard, rerandomization restricts assignment to allocations yielding desirable covariate balance, quantitatively measured by the Mahalanobis distance in covariate means. This methodology transposes the efficiency benefits achieved for mean (average treatment effect, ATE) estimation to the quantile regime, providing inferential methods that address both finite-sample and asymptotic concerns and accommodating the distinctive, non-Gaussian limiting distributional properties that arise for quantile estimators under rerandomization (Han et al., 18 Jan 2026, Wang et al., 2024).

1. Formal Framework and Rerandomization Criterion

The finite-population potential outcomes framework considers $n$ units, each with covariates $X_i\in\mathbb{R}^{K_n}$ and fixed potential outcomes $Y_i(1)$ and $Y_i(0)$. Treatment assignment $Z\in\{0,1\}^n$ is generated such that unit $i$ receives treatment if $Z_i=1$; the observed outcome is $Y_i = Z_iY_i(1) + (1-Z_i)Y_i(0)$.

The QTE estimand at quantile $\alpha$ is the difference in empirical $\alpha$-quantiles,

$$\tau_\alpha = q_{1,\alpha} - q_{0,\alpha},\qquad q_{z,\alpha} = \inf\left\{q : F_z(q)\ge\alpha\right\},$$

where $F_z(q) = \frac{1}{n}\sum_{i=1}^n 1\{Y_i(z)\le q\}$ is the potential outcome CDF under arm $z\in\{0,1\}$.

Rerandomization (ReM) enforces balance by repeatedly sampling treatment assignments until the Mahalanobis distance between treated and control covariate means, $M = nr_1r_0\hat\tau_x^\top S_{xx}^{-1}\hat\tau_x$, satisfies $M\leq a_n$, where $r_z = n_z/n$ and $S_{xx}$ is the finite-population covariance. The stringency is controlled by the threshold $a_n$, which determines the acceptance probability $p_n = P(\chi^2_{K_n}\leq a_n)$ (Han et al., 18 Jan 2026, Wang et al., 2024).

2. Quantile Estimation and Rerandomization-Adjusted Procedures

For the observed data, the empirical CDFs of outcomes in each arm are

$$\hat F_z(q) = \frac{1}{n_z}\sum_{i:Z_i=z} 1\{Y_i \leq q\}$$

and the plug-in quantile estimator is

$$\hat q_{z,\alpha} = \inf\{q : \hat F_z(q)\geq \alpha\}.$$

Thus, the QTE estimator is

$$\hat\tau_\alpha = \hat q_{1,\alpha} - \hat q_{0,\alpha}.$$

The general M-estimator framework characterizes the estimator as the solution $\hat\theta$ to

$\sum_{i=1}^n \psi(O_i; \theta) = 0$

where, for QTE, $\theta=(q_1, q_0)$ and

$$\psi_i(q_1, q_0) = \bigl(A_i[\alpha - \mathbf{1}\{Y_i\le q_1\}],\; (1-A_i)[\alpha-\mathbf{1}\{Y_i\le q_0\}] \bigr)^\top,$$

with $O_i = (A_i, Y_i, X_i)$. This ensures that $q_1$ and $q_0$ solve for the empirical $\alpha$-quantiles in treated and control groups (Wang et al., 2024).

3. Asymptotic Distribution under Rerandomization

Under complete randomization, $\sqrt{n}(\hat\tau_\alpha - \tau_\alpha)$ is asymptotically normal. In contrast, rerandomization induces a non-Gaussian limiting law. Specifically, under mild regularity and conditioning on $M\le a_n$: $$\sqrt{n}(\hat\tau_\alpha - \tau_\alpha) \xrightarrow{d} \tilde V_{qq}^{1/2} \Bigl(\sqrt{1-\tilde R_q^2}\,\varepsilon + \tilde R_q\,L_{K_n, a_n}\Bigr)$$ where:

• $\varepsilon \sim N(0,1)$, independent of $L_{K_n,a_n}$.
• $L_{K_n,a_n} = D_1\,\mid\, \|D\|^2\le a_n$ for $D\sim N(0,I_{K_n})$.
• $\tilde V_{qq}$ is the asymptotic variance under complete randomization (including density scaling factors).
• $\tilde R_q^2$ is the squared multiple correlation between the QTE estimator and the covariate mean difference (Han et al., 18 Jan 2026).

Therefore, the limiting law is a mixture of normal and truncated normal components ("spike and slab" structure), and is non-Gaussian unless the QTE estimator is uncorrelated with the balanced covariates (Han et al., 18 Jan 2026, Wang et al., 2024).

Augmenting the estimating equation with the rerandomization covariate-mean differences eliminates the non-Gaussian component, restoring exact asymptotic normality (Wang et al., 2024).

4. Variance Estimation and Confidence Intervals

Because some counterfactual (joint potential outcome) covariances are unidentifiable, inference is based on conservative estimators. Denoting sample analogs of variance components as $\hat C_n$, $\hat A_n$, and $\hat B_n$, define

$$\hat C_\alpha = \left[ \hat\tau_\alpha - n^{-1/2}\nu_{1-\alpha/2, K_n, a_n}(\hat A_n, \hat B_n), \; \hat\tau_\alpha + n^{-1/2}\nu_{1-\alpha/2, K_n, a_n}(\hat A_n, \hat B_n) \right]$$

where $\nu_{1-\alpha/2, K_n, a_n}(\hat A_n, \hat B_n)$ is the $(1-\alpha/2)$-quantile of the distribution $$\hat A_n^{1/2}\varepsilon + \hat B_n^{1/2} L_{K_n,a_n}$$. This interval is conservative: $$\liminf P(\tau_\alpha \in \hat C_\alpha) \ge 1-\alpha.$$ Alternatively, augmenting the estimating equations as above permits the standard sandwich variance estimator and usual Wald-type confidence intervals using the post-augmentation asymptotic normality (Han et al., 18 Jan 2026, Wang et al., 2024).

5. Efficiency Gains and Theoretical Comparison

Rerandomization reduces the asymptotic variance of $\hat\tau_\alpha$ compared to complete randomization. The percent reduction in asymptotic sampling variance (PRIASV) is characterized as

$\mathrm{PRIASV} = (1 - v_{K_n, a_n}) \tilde R_q^2,$

with $v_{K_n, a_n} = \Var(L_{K_n, a_n}) = P(\chi^2_{K_n+2}\leq a_n)/P(\chi^2_{K_n}\leq a_n)<1$ (Han et al., 18 Jan 2026). Higher $\tilde R_q^2$ reflects greater prognostic strength of the covariates relative to the QTE; more stringent balance (smaller $a_n$) increases variance reduction. Empirical simulations demonstrate PRIASV values between 10–50%, consistent with theory, and variance reductions up to 45% for $R^2 \approx 0.2$–0.5 (Han et al., 18 Jan 2026, Wang et al., 2024).

6. Simulation Evidence and Empirical Performance

Monte Carlo simulations and real-data applications (e.g., IHDP covariates) validate the theory:

• Variance reductions for $\hat\tau_\alpha$ are proportional to PRIASV.
• Shorter confidence intervals (10–30% reduction in length) are achieved with maintained or enhanced nominal coverage.
• Coverage robustness persists at moderate sample sizes ($n\approx 1000$) and allocation imbalance.
• When key baseline covariates are balanced, the realized variance $\widehat V_\text{reduced}$ closely follows $1-R^2$ times the variance under complete randomization, with $R^2$ quantifying the predictive power of rerandomized covariates for the QTE estimator's influence function (Han et al., 18 Jan 2026, Wang et al., 2024).

7. Implementation Guidance and Practical Considerations

Recommended practices include:

• Select rerandomization covariates ($X^r$) with high predictive power for outcome quantiles or the QTE influence function.
• Use the rerandomization (or stratified rerandomization) procedure: iteratively accept treatment assignments only if the sample Mahalanobis distance for $X^r$ is below the prespecified threshold.
• Estimate QTE by fitting quantile regressions or direct sample quantiles in treated and control arms.
• Estimate variances using either the conservative variance estimator (from the non-Gaussian limit) or, after augmenting the estimating equation with covariate-mean imbalances, the usual sandwich estimator.
• For inference, use either the conservative interval in the rerandomization framework or standard Wald intervals post-augmentation (Han et al., 18 Jan 2026, Wang et al., 2024).

In summary, rerandomization for QTE estimation extends the efficiency and robustness gains previously established for average treatment effect settings, introduces unique non-Gaussian inferential phenomena, and is supported by both theoretical and empirical evidence for substantial gains in estimation precision and interval validity in practical applications (Han et al., 18 Jan 2026, Wang et al., 2024).