Quantile regression with measurement errors

2026-06-30

AI Summary

This study addresses the challenge of obtaining consistent estimators in quantile regression when covariates are subject to normal measurement error, a setting complicated by the discontinuity and nonlinearity of the check loss function. The authors propose a novel estimation approach applicable to both linear and nonlinear models, which employs kernel smoothing to handle discontinuities and leverages complex-domain extensions together with moment-generating functions to manage nonlinearity—without requiring joint modeling across multiple quantiles. Within a general quantile regression framework, this work establishes, for the first time, an estimator that achieves root-n consistency and asymptotic normality. Theoretical analysis confirms the standard convergence rate, while numerical simulations and an empirical application to the 2024 Japanese cherry blossom bloom dates demonstrate the method’s practical effectiveness.

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.

Problem

• quantile regression
• measurement errors
• covariates
• normal measurement errors
• consistent estimation

Innovation

• quantile regression
• measurement errors
• kernel smoothing
• moment generating function
• root-n consistency