Advancing multiplicative distortion models in nonlinear regression: relaxing independence and enhancing inference

This paper tackles the intricate challenges posed by multiplicative distortion measurement errors in nonlinear regression models. By eschewing the conventional independence assumption between the confounding variable and unobserved variables, we instead adopt constant conditional mean conditions, thereby broadening the theoretical and practical scope of multiplicative distortion models. Under these refined assumptions, we establish the feasibility of relaxing the independence condition and rigorously investigate the asymptotic properties of the proposed estimators and test statistics, independent of the independence assumption. Our theoretical advancements significantly extend the applicability of conditional mean calibration techniques, encompassing robust confidence interval construction and hypothesis testing for model parameters. To further enhance estimation precision, we introduce a sophisticated second-order nonlinear least squares method. Additionally, we propose a residual-based empirical process test statistic for comprehensive model validation. Simulation studies corroborate the efficacy of our methodology, and a real-world application is analysed to underscore its practical utility and versatility.