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August 2009 Covariate-adjusted nonlinear regression
Xia Cui, Wensheng Guo, Lu Lin, Lixing Zhu
Ann. Statist. 37(4): 1839-1870 (August 2009). DOI: 10.1214/08-AOS627

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.

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Xia Cui. Wensheng Guo. Lu Lin. Lixing Zhu. "Covariate-adjusted nonlinear regression." Ann. Statist. 37 (4) 1839 - 1870, August 2009. https://doi.org/10.1214/08-AOS627

Information

Published: August 2009
First available in Project Euclid: 18 June 2009

zbMATH: 1168.62035
MathSciNet: MR2533473
Digital Object Identifier: 10.1214/08-AOS627

Subjects:
Primary: 62G08 , 62G20 , 62J05

Keywords: asymptotic behavior , confidence region , covariate-adjusted regression , empirical likelihood , Kernel estimation , nonlinear least squares

Rights: Copyright © 2009 Institute of Mathematical Statistics

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Vol.37 • No. 4 • August 2009
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