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This paper considers nonlinear regression models when neither the response variable nor the covariates can be directly observed, but are measured with multiplicative distortion measurement errors. We proposed the second order least squares estimator by including in the criterion function the distance of both the response variable to its conditional mean and the squared response variable to its second conditional moment given the predictor variables. The asymptotic results of the second order nonlinear least squares estimators for the regression parameter are derived. As a by-product, we construct confidence intervals by estimating the asymptotic covariance matrices. We also consider to use three existing calibration procedures to calibrate the unknown response and covariates, namely, the conditional absolute mean calibration, the conditional variance calibration and the logarithmic calibration. The simulation studies shows that these second order nonlinear least squares estimators with three calibration procedures are all superior than the ordinary nonlinear least squares estimators.

Nonlinear multiplicative distortion regression models with second-order estimation