Covariate-adjusted nonlinear regression

TL;DR

This paper introduces a covariate-adjusted nonlinear regression model that accounts for multiplicative distortions in response and predictors, providing consistent estimators and efficient confidence regions through empirical likelihood methods.

Contribution

It develops a novel approach for nonlinear regression with multiplicative distortions, including nonparametric estimation of distorting functions and empirical likelihood-based inference.

Findings

Root n-consistency and asymptotic normality of estimators

Empirical likelihood confidence regions are accurate and self-scale invariant

Application to medical data demonstrates practical utility

Abstract

In this paper, we propose a covariate-adjusted nonlinear regression model. In this model, both the response and predictors can only be observed after being distorted by some multiplicative factors. Because of nonlinearity, existing methods for the linear setting cannot be directly employed. To attack this problem, we propose estimating the distorting functions by nonparametrically regressing the predictors and response on the distorting covariate; then, nonlinear least squares estimators for the parameters are obtained using the estimated response and predictors. Root $n$-consistency and asymptotic normality are established. However, the limiting variance has a very complex structure with several unknown components, and confidence regions based on normal approximation are not efficient. Empirical likelihood-based confidence regions are proposed, and their accuracy is also verified due to its self-scale invariance. Furthermore, unlike the common results derived from the profile methods, even when plug-in estimates are used for the infinite-dimensional nuisance parameters (distorting functions), the limit of empirical likelihood ratio is still chi-squared distributed. This property eases the construction of the empirical likelihood-based confidence regions. A simulation study is carried out to assess the finite sample performance of the proposed estimators and confidence regions. We apply our method to study the relationship between glomerular filtration rate and serum creatinine.