Nonlinear measurement errors models subject to partial linear additive distortion

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.