A bootstrap test for exploring the impact of the covariates in the linear model with non-normal errors

Linear regression models have been extensively studied and used under the assumption that the model error term is normally distributed. However, in many practiced situations, this assumption may not be met. Thus, conclusions of statistical inference and hypothesis testing based on Ordinary Least Square (short for OLS) may be incorrect and unbelievable in condition of error term following non-normal distributions. Based on the above considerations, it is necessary for investigating linear regression models with non-normal errors. In this paper, we focus on the estimation and inference of the linear model with non-normal errors. Specifically, a non-parametric procedure is first developed for calibrating the model. Then, a bootstrap test is presented to detect the significance of the impact of certain covariates on the response based on the idea of variance analysis. Simulation results show that the estimation and proposed test method performs well.

线性回归模型已被广泛研究与应用，其默认前提为模型误差项服从正态分布。然而在诸多实际场景中，该假设往往难以满足。因此，当误差项服从非正态分布时，基于普通最小二乘法（Ordinary Least Square, OLS）得到的统计推断与假设检验结论，可能存在偏差且缺乏可信度。基于上述考量，对带有非正态误差项的线性回归模型开展研究具备重要意义。本文聚焦于带有非正态误差项的线性模型的估计与推断问题。具体而言，本文首先提出一种非参数（non-parametric）流程以完成模型校准。随后，基于方差分析的思想，本文提出一种bootstrap检验（bootstrap test）方法，用以判别特定协变量对响应变量的影响是否显著。仿真实验结果表明，本文所提出的估计方法与检验方法均具备良好的性能表现。