In this paper, we study partially varying-coefficient single-index model where both the response and predictors are observed with multiplicative distortions which depend on a commonly observable confounding variable. Due to the existence of measurement errors, the existing methods cannot be directly applied, so we recommend using the nonparametric regression to estimate the distortion functions and obtain the calibrated variables accordingly. With these corrected variables, the initial estimators of unknown coefficient and link functions are estimated by assuming that the parameter vector \(\beta \) is known. Furthermore, we can obtain the least square estimators of unknown parameters. Moreover, we establish the asymptotic properties of the proposed estimators. Simulation studies and real data analysis are given to illustrate the advantage of our proposed method.

在本文中，我们研究了部分变化系数单指数模型，在该模型中，响应和预测变量均通过乘法失真来观察，这取决于通常可观察的混杂变量。由于测量误差的存在，现有方法不能直接应用，因此我们建议使用非参数回归来估计失真函数并相应地获得校准变量。有了这些修正后的变量，未知系数和链接函数的初始估计量通过假设参数向量\(\beta \)是已知的。此外，我们可以获得未知参数的最小二乘估计量。此外，我们建立了所提出的估计量的渐近特性。给出了模拟研究和实际数据分析来说明我们提出的方法的优点。