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.

本论文针对非线性回归模型（nonlinear regression models）中乘法失真测量误差（multiplicative distortion measurement errors）带来的复杂挑战展开研究。本文摒弃传统将混杂变量（confounding variable）与未观测变量（unobserved variables）视作独立的常规假设，转而采用常数条件均值条件（constant conditional mean conditions），进而拓宽了乘法失真模型的理论与应用边界。在这些精细化的假设框架下，本文论证了放宽独立性条件的可行性，并严格推导了所提估计量（estimators）与检验统计量（test statistics）的渐近性质（asymptotic properties），且该推导不依赖独立性假设。本理论进展显著拓展了条件均值校准技术（conditional mean calibration techniques）的适用范畴，涵盖了模型参数的稳健置信区间构建与假设检验（hypothesis testing）流程。为进一步提升估计精度，本文提出了一种精巧的二阶非线性最小二乘法（second-order nonlinear least squares method）。此外，本文还设计了用于开展全面模型验证的基于残差的经验过程检验统计量（residual-based empirical process test statistic）。模拟研究（simulation studies）证实了所提方法的有效性，同时通过一项实际应用（real-world application）案例分析，凸显了该方法的实用价值与通用性。