Nonlinear measurement error models subject to additive distortion

We study nonlinear regression models when the response and predictors are unobservable and distorted in a multiplicative fashion by additive models of some observed confounding variables. After approximating the additive nonparametric components via polynomial splines and calibrating the error-prone response and predictors, we develop a semi-parametric profile nonlinear least squares procedure to estimate the parameters of interest. We show that the resulting estimators are asymptotically normal. We further suggest an empirical likelihood-based statistic for statistical inference to improve the accuracy of the associated normal approximation with the aim to avoid estimating the asymptotic covariance matrix that involves infinite-dimensional nuisance of additive distorting functions. We also show that the empirical likelihood statistic is asymptotically chi-squared. Moreover, a test procedure is proposed to check whether the parametric model is adequate or not under this distorted measurement error setting. A wild bootstrap procedure is suggested to compute p-values. Simulation studies are conducted to examine the performance of the proposed procedures. The methods are applied to analyze real data from a low birth infants weight for an illustration.