In this paper, we introduce the mixed estimators based on product least relative error estimation and least squares estimation in a multiplicative linear regression model. The asymptotic properties for the mixed estimators are established. We present some explicit expressions of the optimal estimator of the mixed estimators, and we also suggest some numerical solutions in the simulation studies and real data analysis. Studying model checking problems for multiplicative linear regression models, we propose four test statistics. One is the score-type test statistic, the second one is the residual-based empirical process test statistic marked by proper functions of the covariates. The third one is the integrated conditional moment test statistic by using linear projection weighting function, and the fourth one is the adaptive model test statistic. These test statistics are all related to the mixed estimators. The asymptotic properties of these test statistics are established, and some bootstrap procedures for calculating the critical values are also proposed. Simulation studies are conducted to demonstrate the performance of the proposed estimation procedures, and a real example is analyzed to illustrate its practical usage.

中文翻译：

---

具有混合估计量的乘法线性回归模型的模型检查

在本文中，我们在乘法线性回归模型中引入了基于乘积最小相对误差估计和最小二乘估计的混合估计器。建立了混合估计量的渐近性质。我们提出了混合估计量的最优估计量的一些显式表达式，并且我们还在模拟研究和实际数据分析中提出了一些数值解。研究乘法线性回归模型的模型检查问题，我们提出了四个检验统计量。一是分数型检验统计量，二是以协变量的适当函数为标志的基于残差的经验过程检验统计量。第三个是利用线性投影加权函数的综合条件矩检验统计量，第四个是自适应模型检验统计量。这些检验统计量都与混合估计量有关。建立了这些检验统计量的渐近性质，并提出了一些计算临界值的引导程序。进行了模拟研究以证明所提出的估计程序的性能，并分析了一个真实的例子来说明其实际用途。