L_{1}-Estimation for covariate-adjusted regression

Yaodong Sun,Dehui Wang

doi:10.1186/s13660-020-02324-w

Abstract

We study a covariate-adjusted regression(CAR) model that is proposed for such situations where both predictors and response in a regression model are not directly observable but are distorted by a multiplicative factor that is determined by an unknown function of some observable covariate. By establishing a connection to varying-coefficient models, we present the local linear L_{1}-estimation method when the underlying error distribution deviates from a normal distribution. The robust estimators of parameters are proposed in the underlying regression model. The consistency and asymptotic normality of the robust estimators are investigated. Since the limit distribution depends on the unknown components of the errors, an empirical likelihood ratio method based on L_{1} estimator is proposed. The confidence intervals for the regression coefficients are constructed. Simulation results demonstrate the superiority of the proposed estimators over other classical estimators when the underlying errors have heavy tails. Pima Indian diabetes data set is conducted to illustrate the performance of the proposed method, where the response and predictors are potentially contaminated by body mass index.

Highlights

Covariate-adjusted regression(CAR) was initially proposed for regression analysis by Sentürk and Müller [18], where both the response and predictors are not directly observed
The available data are distorted by unknown functions of some common observable covariate
An example is the fibrinogen data collected on 69 hemodialysis patients, where the regression of fibrinogen level on serum transferrin level is of interest in Kaysen et al [11]

Summary

Introduction

Covariate-adjusted regression(CAR) was initially proposed for regression analysis by Sentürk and Müller [18], where both the response and predictors are not directly observed. Sentürk and Müller [18] suggested that the confounding variable affects the primary variables through a flexible multiplicative unknown function. Such way of adjustment may reduce non-negligible bias and lead to consistent estimators of the parameters of interest, which is through dividing by the body mass index identified as a common confounder. An estimated empirical log-likelihood approach to construct the confidence region of the regression parameter is developed. Our goal is to estimate the unknown parameter γ consistently based the observed data, and to further establish asymptotic normality for the proposed estimators. We see that the coverage probabilities increase

50 Normal LP L1

Findings

Discussion