pith. sign in

arxiv: 2607.00188 · v1 · pith:MSUQLBTRnew · submitted 2026-06-30 · 📊 stat.ME

Quantile regression with measurement errors

Mushan Li , Yanyuan Ma , Liqun Wang This is my paper

Pith reviewed 2026-07-02 17:31 UTC · model grok-4.3

classification 📊 stat.ME
keywords quantile regressionmeasurement errornormal errorskernel smoothingmoment generating functionconsistencyasymptotic normalitybias correction
0
0 comments X

The pith

A new estimator using kernel smoothing and complex-domain moment generating functions delivers root-n consistent quantile regression estimates even when covariates have normal measurement errors.

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

The paper constructs an estimator that corrects bias from normal measurement errors in the covariates of a quantile regression model. The approach applies to both linear and nonlinear specifications and avoids any requirement that the model hold simultaneously at several quantile levels. Kernel smoothing addresses the jump discontinuity of the quantile loss, while an extension into the complex plane lets moment generating functions linearize the otherwise intractable nonlinearity. Under standard conditions the estimator attains the usual root-n rate and asymptotic normality, providing the first consistent procedure for this general setting.

Core claim

We devise a novel estimator for a general quantile regression model with normal measurement errors in the covariates. The method is applicable to both linear and nonlinear quantile regressions and does not impose the quantile requirement on multiple quantile levels simultaneously. We circumvent the difficulties caused by discontinuity in quantile regression through kernel smoothing, and overcome the nonlinearity inherent in quantile regression via considering extension to the complex domain and moment generating functions. We show that the resulting estimator achieves the standard root-n consistency and asymptotic normality under mild conditions. This is the first consistent estimator in a g

What carries the argument

Kernel smoothing of the quantile loss combined with complex-domain moment generating functions that correct the measurement-error bias.

If this is right

  • The estimator remains consistent for both linear and nonlinear quantile regression specifications.
  • No simultaneous imposition of the model across multiple quantile levels is required.
  • Standard root-n consistency and asymptotic normality hold under mild regularity conditions.
  • The procedure is illustrated on simulated data and on a real dataset of Cherry Blossom flowering times.

Where Pith is reading between the lines

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

  • Analyses that previously discarded quantile regression because of suspected covariate measurement error can now be revisited with a consistent method.
  • The separation of discontinuity handling from nonlinearity correction may allow the same machinery to be reused for other non-smooth or nonlinear estimating equations.
  • Empirical work in fields that routinely encounter imprecise covariates can adopt the estimator without needing to collect error-free data.

Load-bearing premise

The measurement errors in the covariates are normally distributed.

What would settle it

A large-sample simulation or theoretical calculation in which the proposed estimator fails to converge at the root-n rate when the measurement errors are exactly normal would falsify the consistency claim.

Figures

Figures reproduced from arXiv: 2607.00188 by Liqun Wang, Mushan Li, Yanyuan Ma.

Figure 1
Figure 1. Figure 1: Comparison of different methods. ✸ and lines represent the estimator and its 95% confidence interval. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
read the original abstract

We devise a novel estimator for a general quantile regression model with normal measurement errors in the covariates. The method is applicable to both linear and nonlinear quantile regressions and does not impose the quantile requirement on multiple quantile levels simultaneously. We circumvent the difficulties caused by discontinuity in quantile regression through kernel smoothing, and overcome the nonlinearity inherent in quantile regression via considering extension to the complex domain and moment generating functions. We show that the resulting estimator achieves the standard root-$n$ consistency and asymptotic normality under mild conditions. The performance of the proposed method is illustrated via numerical simulations and a real data example related to Cherry Blossom times in Japan in 2024. This is the first consistent estimator in a general quantile regression problem with normal measurement errors.

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 a novel estimator for general quantile regression (linear and nonlinear) with normal measurement errors in covariates. It combines kernel smoothing of the check function with a bias-correction step that extends the moment generating function to the complex domain, claiming that the resulting estimator attains root-n consistency and asymptotic normality under mild conditions without requiring the quantile restriction to hold at multiple levels simultaneously. The claims are illustrated with numerical simulations and a real-data application to Cherry Blossom flowering times in Japan.

Significance. If the central theoretical claims are correct, the work would constitute a meaningful methodological advance by supplying the first consistent estimator for this class of problems; the combination of kernel smoothing and complex-domain MGF correction is a technically interesting route around the non-differentiability and nonlinearity obstacles that have previously blocked progress.

major comments (2)
  1. [Abstract and theoretical development section] Abstract and theoretical development section: the manuscript asserts root-n consistency and asymptotic normality but supplies neither the derivation steps nor the regularity conditions under which they hold; because this is the load-bearing claim of the paper, the absence of any proof or verification sketch prevents assessment of whether the estimator actually achieves the stated rate.
  2. [Abstract and theoretical development section] The normality assumption on the measurement errors is essential to the MGF correction step, yet the paper does not discuss the sensitivity of the procedure to departures from normality or provide any diagnostic for when the assumption is plausible.
minor comments (2)
  1. The real-data example would benefit from an explicit statement of how the measurement-error variance was estimated or assumed.
  2. Notation for the complex extension of the MGF is introduced without a dedicated display equation, making it difficult to follow the bias-correction algebra.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the two major comments. We address each below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract and theoretical development section] Abstract and theoretical development section: the manuscript asserts root-n consistency and asymptotic normality but supplies neither the derivation steps nor the regularity conditions under which they hold; because this is the load-bearing claim of the paper, the absence of any proof or verification sketch prevents assessment of whether the estimator actually achieves the stated rate.

    Authors: The main text states the root-n consistency and asymptotic normality result together with the claim that it holds under mild conditions, but we accept that a self-contained proof sketch and an explicit list of regularity conditions are needed for readers to evaluate the claim. In the revision we will insert a concise proof outline (with key steps) and the full set of regularity conditions into the theoretical development section. revision: yes

  2. Referee: [Abstract and theoretical development section] The normality assumption on the measurement errors is essential to the MGF correction step, yet the paper does not discuss the sensitivity of the procedure to departures from normality or provide any diagnostic for when the assumption is plausible.

    Authors: We agree that the normality assumption is central to the complex-domain MGF correction and that its sensitivity should be examined. In the revised manuscript we will add a dedicated subsection that (i) analytically indicates where normality enters the bias-correction step, (ii) reports simulation results under moderate departures from normality, and (iii) suggests simple diagnostics (e.g., normality checks on replicate measurements when available). revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation relies on kernel smoothing of the check function combined with an MGF-based bias correction that exploits the explicit normality assumption on measurement errors. The root-n consistency and asymptotic normality are obtained via standard asymptotic arguments under stated mild conditions; these are not shown to reduce by construction to fitted parameters or prior self-citations. The normality assumption is presented as an input that enables the MGF step rather than being derived from the target result. No self-definitional, fitted-input, or self-citation-load-bearing reductions are exhibited in the provided abstract or high-level description.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities listed. The approach relies on standard kernel smoothing and MGF properties for normal errors, but these are not detailed.

pith-pipeline@v0.9.1-grok · 5642 in / 1078 out tokens · 22712 ms · 2026-07-02T17:31:44.000374+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

  1. [1]

    Journal of the American Statistical Association , volume=

    Simulation-extrapolation: the measurement error jackknife , author=. Journal of the American Statistical Association , volume=. 1995 , publisher=

    work page 1995

  2. [2]

    Journal of the American Statistical association , volume=

    Simulation-extrapolation estimation in parametric measurement error models , author=. Journal of the American Statistical association , volume=. 1994 , publisher=

    work page 1994

  3. [3]

    Econometrica , volume=

    Errors in the dependent variable of quantile regression models , author=. Econometrica , volume=. 2021 , publisher=

    work page 2021

  4. [4]

    Journal of econometrics , volume=

    Measurement errors in quantile regression models , author=. Journal of econometrics , volume=. 2017 , publisher=

    work page 2017

  5. [5]

    Journal of the American Statistical Association , volume=

    Using SIMEX for smoothing-parameter choice in errors-in-variables problems , author=. Journal of the American Statistical Association , volume=. 2008 , publisher=

    work page 2008

  6. [6]

    kneedle

    Finding a" kneedle" in a haystack: Detecting knee points in system behavior , author=. 2011 31st international conference on distributed computing systems workshops , pages=. 2011 , organization=

    work page 2011

  7. [7]

    Statistica Sinica , volume=

    Partially functional linear quantile regression with measurement errors , author=. Statistica Sinica , volume=

    work page

  8. [8]

    2012 , journal=

    Analysis on Censored Quantile Residual Life Model via Spline Smoothing , author=. 2012 , journal=

    work page 2012

  9. [9]

    Electronic Journal of Statistics , volume=

    Quantile regression for interval censored data using an Enriched Laplace distribution , author=. Electronic Journal of Statistics , volume=. 2025 , publisher=

    work page 2025

  10. [10]

    Lifetime Data Analysis , pages=

    Quantile regression under dependent censoring with unknown association , author=. Lifetime Data Analysis , pages=. 2025 , publisher=

    work page 2025

  11. [11]

    Journal of the American Statistical Association , volume=

    An adapted loss function for censored quantile regression , author=. Journal of the American Statistical Association , volume=. 2019 , publisher=

    work page 2019

  12. [12]

    Biometrika , volume=

    Wild residual bootstrap inference for penalized quantile regression with heteroscedastic errors , author=. Biometrika , volume=. 2018 , publisher=

    work page 2018

  13. [13]

    Chinese Journal of Applied Probability and Statistics , volume=

    Instrumental variable estimation in linear quantile regression models with measurement error , author=. Chinese Journal of Applied Probability and Statistics , volume=

    work page

  14. [14]

    Metrika , volume=

    Composite quantile regression estimation of linear error-in-variable models using instrumental variables , author=. Metrika , volume=. 2020 , publisher=

    work page 2020

  15. [15]

    Econometrica: journal of the Econometric Society , pages=

    Regression quantiles , author=. Econometrica: journal of the Econometric Society , pages=. 1978 , publisher=

    work page 1978

  16. [16]

    Annual review of economics , volume=

    Quantile regression: 40 years on , author=. Annual review of economics , volume=. 2017 , publisher=

    work page 2017

  17. [17]

    Journal of the Royal Statistical Society Series D: The Statistician , volume=

    Quantile regression: applications and current research areas , author=. Journal of the Royal Statistical Society Series D: The Statistician , volume=. 2003 , publisher=

    work page 2003

  18. [18]

    2017 , publisher=

    Handbook of Quantile Regression , author=. 2017 , publisher=

    work page 2017

  19. [19]

    2019 , publisher=

    Li, Mushan and Tu, Shiqi and Li, Zijia and Tan, Fengxiang and Liu, Jian and Wang, Qian and Zhang, Yuannyu and Xu, Jian and Zhang, Yijing and Zhou, Feng and others , journal=. 2019 , publisher=

    work page 2019

  20. [20]

    Statistics in Medicine , volume=

    Robust estimation of mean-variance relation , author=. Statistics in Medicine , volume=. 2024 , publisher=

    work page 2024

  21. [21]

    Annual Review of Statistics and Its Application , volume=

    An update on measurement error modeling , author=. Annual Review of Statistics and Its Application , volume=. 2024 , publisher=

    work page 2024

  22. [22]

    Journal of multivariate analysis , volume=

    Semiparametric regression for measurement error model with heteroscedastic error , author=. Journal of multivariate analysis , volume=. 2019 , publisher=

    work page 2019

  23. [23]

    2009 , publisher=

    Measurement error models , author=. 2009 , publisher=

    work page 2009

  24. [24]

    Handbook of econometrics , volume=

    Large sample estimation and hypothesis testing , author=. Handbook of econometrics , volume=. 1994 , publisher=

    work page 1994

  25. [25]

    Journal of the American Statistical Association , volume=

    Quantile regression with measurement error , author=. Journal of the American Statistical Association , volume=. 2009 , publisher=

    work page 2009

  26. [26]

    Journal of the American Statistical Association , volume=

    Optimal Rates of Convergence for Deconvolving a Density , author=. Journal of the American Statistical Association , volume=

    work page

  27. [27]

    Biometrika , volume=

    Corrected-loss estimation for quantile regression with covariate measurement errors , author=. Biometrika , volume=

    work page

  28. [28]

    American Journal of Epidemiology , volume=

    Maternal, birth, and early-life influences on adult body size in women , author=. American Journal of Epidemiology , volume=. 2007 , publisher=

    work page 2007

  29. [29]

    Statistica Sinica , pages=

    Quantile regression estimates for a class of linear and partially linear errors-in-variables models , author=. Statistica Sinica , pages=

    work page

  30. [30]

    2004 , journal=

    Locally Efficient Semiparametric Estimators for Functional Measurement Error Models , author=. 2004 , journal=

    work page 2004

  31. [31]

    2006 , journal=

    Closed Form Semiparametric Estimators for Measurement Error Models , author=. 2006 , journal=

    work page 2006

  32. [32]

    2017 , journal=

    Simultaneous treatment of unspecified heteroskedastic model error distribution and mismeasured covariates for restricted moment models , author=. 2017 , journal=

    work page 2017

  33. [33]

    2006 , address=

    Measurement Error in Nonlinear Models: A Modern Perspective , author=. 2006 , address=

    work page 2006

  34. [34]

    2015 , journal=

    Smoothed and Corrected Score Approach to Censored Quantile Regression with Measurement Errors , author=. 2015 , journal=

    work page 2015

  35. [35]

    2011 , journal=

    Censored Quantile Regression with Covariate Measurement Errors , author=. 2011 , journal=

    work page 2011

  36. [36]

    1989 , journal=

    Unbiased Estimation of a Nonlinear Function of a Normal Mean with Application to Measurement Error Models , author=. 1989 , journal=

    work page 1989