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arxiv: 2606.04128 · v1 · pith:ZGFCJ6HRnew · submitted 2026-06-02 · 📊 stat.ME

On prediction-powered inference for quantile regression via convolution smoothing

Shota Takeishi , Jimin Ding , Xuming He This is my paper

Pith reviewed 2026-06-28 08:35 UTC · model grok-4.3

classification 📊 stat.ME
keywords quantile regressionprediction-powered inferenceconvolution smoothingcheck lossasymptotic distributionmodel misspecificationsurrogate outcomesensemble estimator
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The pith

Convolution smoothing of the check-loss objective makes prediction-powered quantile regression computationally tractable while reducing overcoverage in confidence intervals.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

In settings where gold-standard outcomes are scarce but AI surrogate predictions are abundant, extending prediction-powered inference to quantile regression runs into computational trouble from the non-smooth check loss and produces overly wide confidence intervals. The paper introduces convolution smoothing to create differentiable approximations of the check loss and derives two estimator variants. These variants admit tractable optimization and come with asymptotic normality results that continue to hold when the linear quantile model is misspecified. Numerical experiments indicate the smoothed estimators produce narrower intervals that still maintain coverage, and the authors also form an ensemble of the two variants.

Core claim

We propose a convolution-based smoothing of the check-loss objective and develop two variants of the estimator. The proposed estimators are computationally tractable, and our numerical studies show that they mitigate overcoverage. As a theoretical contribution, we establish the asymptotic distributions of the proposed estimators under a possibly misspecified linear quantile regression model. We further propose an ensemble of the two estimators.

What carries the argument

Convolution smoothing of the check-loss objective, which replaces the non-differentiable loss with a smoothed version that supports gradient-based computation while retaining the asymptotic properties needed for valid inference.

If this is right

  • The two smoothed estimators can be computed efficiently even when the surrogate dataset is large.
  • Asymptotic normality supplies explicit confidence intervals whose coverage improves over the naive extension.
  • An ensemble of the two variants can be formed to balance their individual properties.
  • The approach extends directly to applications such as housing price quantiles where limited gold-standard observations are supplemented by predictions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same smoothing idea could be applied to other non-smooth estimating equations that arise in prediction-powered settings.
  • In domains that rely on AI surrogates, such as medical risk scores or financial forecasting, the method may yield tighter quantile estimates without sacrificing validity.
  • Empirical choice of the smoothing bandwidth could be studied to optimize the trade-off between computational speed and interval width.

Load-bearing premise

The convolution smoothing preserves the key properties of the check loss so that the derived asymptotic distributions remain valid and do not introduce bias that invalidates the central limit theorem results, even under model misspecification.

What would settle it

If repeated simulations under the misspecified linear model show that the smoothed estimators fail to achieve the stated asymptotic normality or that their confidence intervals continue to exhibit the same degree of overcoverage as the unsmoothed version, the central claims would be falsified.

Figures

Figures reproduced from arXiv: 2606.04128 by Jimin Ding, Shota Takeishi, Xuming He.

Figure 1
Figure 1. Figure 1: Empirical coverage for 95% confidence intervals across methods. Each point shows the [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Confidence interval length averaged across the coefficients [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: House price versus appraised value [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Point estimates and 95% confidence intervals across quantile levels for the coefficients on [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Undercoverage behavior for the coefficient on [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Sensitivity to hn, averaged across coefficients X1, X2, and X3 (τ = 0.5, large setting): (a) empirical coverage and (b) CI length. known. We further demonstrate the validity of the proposed calibration step through a preliminary simulation study. Define the smoothed score with the calibrated surrogate as Ψˆ hn,Wˆ (β; ˆη) := 1 n Xn i=1 ψhn (β; Yi , Xi) − Wˆ   1 n Xn i=1 ψhn (β; Yˆ i(ˆη), Xi) − 1 N nX +N i… view at source ↗
Figure 7
Figure 7. Figure 7: Empirical coverage for 95% confidence intervals for “LAB” and “SD-OPT” at [PITH_FULL_IMAGE:figures/full_fig_p029_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Confidence interval length averaged across the coefficients [PITH_FULL_IMAGE:figures/full_fig_p032_8.png] view at source ↗
read the original abstract

This paper studies quantile regression in a data-limited setting where the gold-standard outcome is available only for a limited number of observations, whereas a surrogate outcome is widely available. Such settings are becoming increasingly common with the availability of low-cost predictions from modern AI, motivating a growing line of research on "prediction-powered inference," for improved statistical inference. Naively extending this framework to quantile regression, however, raises two challenges: computational difficulties due to the discontinuity of the subgradient, and overly conservative confidence intervals. To address these issues, we propose a convolution-based smoothing of the check-loss objective and develop two variants of the estimator. The proposed estimators are computationally tractable, and our numerical studies show that they mitigate overcoverage. As a theoretical contribution, we establish the asymptotic distributions of the proposed estimators under a possibly misspecified linear quantile regression model. We further propose an ensemble of the two estimators and illustrate the proposed methods through simulations and an application to a local housing dataset.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes convolution-based smoothing of the check-loss objective for quantile regression in the prediction-powered inference setting, where gold-standard outcomes are limited but surrogate predictions are abundant. It develops two variants of the resulting estimator, establishes their asymptotic distributions under a possibly misspecified linear quantile regression model, proposes an ensemble of the two, and reports that numerical studies show mitigation of overcoverage, with an illustration on a local housing dataset.

Significance. If the asymptotic results hold, the work supplies a computationally tractable route to quantile regression under PPI that addresses both non-differentiability and conservative intervals. The simulation evidence on overcoverage reduction and the real-data application constitute concrete strengths.

major comments (2)
  1. [Theoretical contribution (abstract)] Theoretical contribution paragraph: the assertion that the smoothed estimators retain the same asymptotic normal distribution (after centering) as the unsmoothed quantile regression estimator under misspecification requires that the convolution bias term is o_p(n^{-1/2}). The manuscript must specify the bandwidth sequence h_n and kernel conditions that guarantee this rate uniformly in the misspecified regime; without an explicit argument controlling the approximation error, the central CLT claim is not yet load-bearing.
  2. [Numerical studies] Numerical studies section: the reported mitigation of overcoverage is central to the practical claim, yet the abstract supplies no information on how standard errors or coverage probabilities were computed, what data-exclusion rules were applied, or how the bandwidth was selected in finite samples. These details are needed to evaluate whether the simulation design actually supports the overcoverage claim.
minor comments (2)
  1. [Abstract] The abstract refers to 'two variants of the estimator' without naming or briefly characterizing them; adding one-sentence descriptors would improve readability.
  2. [Methods] Notation for the convolution kernel and bandwidth should be introduced consistently when first used in the methods section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below and indicate where revisions will be made.

read point-by-point responses

  1. Referee: [Theoretical contribution (abstract)] Theoretical contribution paragraph: the assertion that the smoothed estimators retain the same asymptotic normal distribution (after centering) as the unsmoothed quantile regression estimator under misspecification requires that the convolution bias term is o_p(n^{-1/2}). The manuscript must specify the bandwidth sequence h_n and kernel conditions that guarantee this rate uniformly in the misspecified regime; without an explicit argument controlling the approximation error, the central CLT claim is not yet load-bearing.

    Authors: We agree that an explicit statement of the bandwidth and kernel conditions is necessary to rigorously justify that the smoothing bias is o_p(n^{-1/2}) uniformly under misspecification. The current derivation assumes this rate but does not list the precise requirements (e.g., h_n = o(n^{-1/4}) with a second-order kernel). We will add these conditions and the corresponding approximation-error bound to the theoretical section in the revision. revision: yes

  2. Referee: [Numerical studies] Numerical studies section: the reported mitigation of overcoverage is central to the practical claim, yet the abstract supplies no information on how standard errors or coverage probabilities were computed, what data-exclusion rules were applied, or how the bandwidth was selected in finite samples. These details are needed to evaluate whether the simulation design actually supports the overcoverage claim.

    Authors: The computational details for standard errors (via the asymptotic variance formula), coverage evaluation, data-exclusion rules, and bandwidth selection (cross-validation) are fully specified in Section 4. To improve transparency, we will revise the abstract to include a short clause noting that these quantities follow the procedures described in the numerical studies section. revision: partial

Circularity Check

0 steps flagged

No circularity; asymptotics and estimators derived independently

full rationale

The paper introduces convolution smoothing of the check-loss for quantile regression under prediction-powered inference with limited gold-standard outcomes. It develops two estimator variants, reports numerical mitigation of overcoverage, and states that asymptotic distributions are established under possible misspecification of the linear quantile model. No equations, definitions, or claims in the abstract or description reduce the target asymptotic result to a fitted parameter by construction, nor do they rely on self-citation chains or imported uniqueness theorems. The derivation chain appears self-contained against external benchmarks, with smoothing introduced as a computational device whose bias control is asserted via separate theoretical arguments rather than by redefinition of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; ledger left empty pending full text.

pith-pipeline@v0.9.1-grok · 5693 in / 1073 out tokens · 16510 ms · 2026-06-28T08:35:35.307157+00:00 · methodology

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Reference graph

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