Quantile Regression Estimates for a Class of Linear and Partially Linear Errors-in-Variables Models

Abstract

We consider the problem of estimating quantile regression coefficients in errors-in-variables models. When the error variables for both the response and the manifest variables have a joint distribution that is spherically symmetric but is otherwise unknown, the regression quantile estimates based on orthogonal residuals are shown to be consistent and asymptotically normal. We also extend the work to partially linear models when the response is related to some additional covariate. Regression analysis is routinely carried out in all areas of statistical appli- cations to explain how a dependent variable Y relates to independent variables X. Most authors consider the estimation or inference problems based on data observed on both X and Y variables. However, the covariates are not always observable without error. If X is observed subject to random error, the regres- sion model is usually called the errors-in-variables (EV) model. A careful study of such models is often needed, as the standard results on regression models do not carry over. The best-known is the effect of attenuation for the likelihood- based estimators without correction for the measurement error in X. A detailed coverage of linear errors-in-variables models can be found in Fuller (1987). More recent work on nonlinear models with measurement errors can be found in Car- roll, Ruppert and Stefanski (1995). The literature on EV models are mainly confined to estimating the conditional mean function of Y given X, assuming Gaussian errors. In the present paper we attempt to consider conditional me- dian and other quantile functions, as pioneered by Koenker and Bassett (1978), for a class of unspecified error distributions. For the usefulness of conditional quantiles, see examples and discussions found in Efron (1991) and He (1997), among many others. Let us start with the EV model Yi = X T