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February 2018 Nonlinear measurement errors models subject to partial linear additive distortion
Jun Zhang, Nanguang Zhou, Qian Chen, Tianyue Chu
Braz. J. Probab. Stat. 32(1): 86-116 (February 2018). DOI: 10.1214/16-BJPS333

Abstract

We study nonlinear regression models when the response and predictors are unobservable and distorted in a multiplicative fashion by partial linear additive models (PLAM) of some observed confounding variables. After approximating the additive nonparametric components in the PLAM via polynomial splines and calibrating the unobserved response and unobserved predictors, we develop a semi-parametric profile nonlinear least squares procedure to estimate the parameters of interest. The resulting estimators are shown to be asymptotically normal. To construct confidence intervals for the parameters of interest, an empirical likelihood-based statistic is proposed to improve the accuracy of the associated normal approximation. We also show that the empirical likelihood statistic is asymptotically chi-squared. Moreover, a test procedure based on the empirical process is proposed to check whether the parametric regression model is adequate or not. A wild bootstrap procedure is proposed to compute $p$-values. Simulation studies are conducted to examine the performance of the estimation and testing procedures. The methods are applied to re-analyze real data from a diabetes study for an illustration.

Citation

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Jun Zhang. Nanguang Zhou. Qian Chen. Tianyue Chu. "Nonlinear measurement errors models subject to partial linear additive distortion." Braz. J. Probab. Stat. 32 (1) 86 - 116, February 2018. https://doi.org/10.1214/16-BJPS333

Information

Received: 1 April 2015; Accepted: 1 August 2016; Published: February 2018
First available in Project Euclid: 3 March 2018

zbMATH: 06973950
MathSciNet: MR3770865
Digital Object Identifier: 10.1214/16-BJPS333

Keywords: Bootstrap approximation , confounding variables , covariate-adjusted regression , distorting function , empirical likelihood , empirical process , measurement errors models , model checking , multiplicative effect , regression spline

Rights: Copyright © 2018 Brazilian Statistical Association

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Vol.32 • No. 1 • February 2018
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