Corrected-loss estimation for quantile regression with covariate measurement errors

Abstract

We study estimation in quantile regression when covariates are measured with errors. Existing methods require stringent assumptions, such as spherically symmetric joint distribution of the regression and measurement error variables, or linearity of all quantile functions, which restrict model flexibility and complicate computation. In this paper, we develop a new estimation approach based on corrected scores to account for a class of covariate measurement errors in quantile regression. The proposed method is simple to implement. Its validity requires only linearity of the particular quantile function of interest, and it requires no parametric assumptions on the regression error distributions. Finite-sample results demonstrate that the proposed estimators are more efficient than the existing methods in various models considered.

t-Type corrected-loss estimation for error-in-variable model

“…Another limitation of the paper is the Laplacian assumption for the measurement error u . If u follows a normal distribution, then we can also derive a new version of objective function just following the ideas in Wang et al (2012). But the objective function is very different from the one in this paper and it involves imaginary parts, so we decide to omit it.…”

“…As Wang et al (2012) mentioned, the Laplace distribution is often used for modelling data with tails heavier than normal. We also refer to Stefanski and Carroll (1990), Hong and Tamer (2003), Richardson and Hollinger (2005), Purdom and Holmes (2005), Visscher (2006) and McKenzie et al (2008) for discussions of Laplace measurement errors.…”

Partially linear support vector orthogonal quantile regression with measurement errors

“…The covariates of interest are often not observable and instead are measured with errors. Disregarding these measurement errors often leads to bias in estimating the mean and quantile functions (Fuller, 1987;Carroll et al, 2006;Wei and Carroll, 2009;Ma and Yin, 2011;Wang et al, 2012). This is the motivation for investigating measurement error models, which are frequently encountered by researchers conducting empirical studies in the social and natural sciences.…”

“…During the last two decades, partially linear measurement errors models have received much attention in the literature as a generalization of the linear measurement errors model. However, less attention has been paid to quantile regression (QR) than to mean regression with covariate measurement errors because of two main difficulties for correcting the bias in QR caused by measurement error (Wang et al, 2012). One is that a parametric regression-error likelihood is usually not specified in QR.…”

“…There have been some works on measurement error in QR (He and Liang, 2000;Chesher, 2001;Schennach, 2008;Wei and Carroll, 2009;Wang et al, 2012;Ma and Yin, 2011). In this paper we propose the partially linear support vector orthogonal QR (PLSVOQR) model with covariate measurement errors by applying quantile loss function of orthogonal residuals to the formulation of semiparametric support vector QR (SSVQR) of Shim et al (2012).…”