Estimation via corrected scores in general semiparametric regression models with error-prone covariates

Arnab Maity,Tatiyana V Apanasovich

doi:10.1214/11-ejs647

Abstract

This paper considers the problem of estimation in a general semiparametric regression model when error-prone covariates are modeled parametrically while covariates measured without error are modeled nonparametrically. To account for the effects of measurement error, we apply a correction to a criterion function. The specific form of the correction proposed allows Monte Carlo simulations in problems for which the direct calculation of a corrected criterion is difficult. Therefore, in contrast to methods that require solving integral equations of possibly multiple dimensions, as in the case of multiple error-prone covariates, we propose methodology which offers a simple implementation. The resulting methods are functional, they make no assumptions about the distribution of the mismeasured covariates. We utilize profile kernel and backfitting estimation methods and derive the asymptotic distribution of the resulting estimators. Through numerical studies we demonstrate the applicability of proposed methods to Poisson, logistic and multivariate Gaussian partially linear models. We show that the performance of our methods is similar to a computationally demanding alternative. Finally, we demonstrate the practical value of our methods when applied to Nevada Test Site (NTS) Thyroid Disease Study data.

Highlights

Regression models with measurement errors arise frequently in practice and have attracted much attention in the statistical literature
Semiparametric regression models with errors in covariates have been considered by several authors in the attempt to develop measurement error calibration techniques when the errors are in the linear part of linear regression ([9]) or generalized linear regression ([10]) models. [30] used a method of moments and deconvolution to construct the calibration for the case of partially linear models when the mismeasured covariate appears in parametric and nonparametric parts
We consider the problem of estimation in a general semiparametric regression model when error-prone covariates are modeled parametrically while covariates measured without error are modeled nonparametrically

Summary

Introduction

Regression models with measurement errors arise frequently in practice and have attracted much attention in the statistical literature. Semiparametric regression models with errors in covariates have been considered by several authors in the attempt to develop measurement error calibration techniques when the errors are in the linear part of linear regression ([9]) or generalized linear regression ([10]) models. [30] used a method of moments and deconvolution to construct the calibration for the case of partially linear models when the mismeasured covariate appears in parametric and nonparametric parts. All of the above methodologies take advantage of the fact that unknown parameters in a parametric part enter the model through a linear combination with error prone covariates. Maity and Apanasovich general semiparametric regression problem where parameters can enter the model through any known function of covariates. Even though SIMEX is a general-purpose, widely applicable method for correcting parameter estimates for the biases induced by measurement error in covariates, it suffers from relying on a rather heuristic extrapolation step ([4])

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