狭义相对论下的线性回归

摘要：

本文研究了在狭义相对论的特定背景下，用普通最小二乘法（OLS）估计简单线性回归的参数所带来的问题，其中自变量被限制在一个开区间（-c，c）内。研究发现，斜率系数的OLS估计在洛仑兹速度变换下不是不变的。在此基础上，提出了狭义相对论下线性回归参数的另一种估计。该估计量可以被认为是狭义相对论下OLS估计量的推广;当c趋于无穷大时，该估计量及其方差分别收敛于OLS估计量及其方差。该估计量的方差大于OLS估计量的方差，这意味着在狭义相对论下，使用OLS估计量及其方差进行假设检验可能导致宽松的检验.

英文摘要：

This study investigated the problem posed by using ordinary least squares (OLS) to estimate parameters of simple linear regression under a specific context of special relativity, where an independent variable is restricted to an open interval, (-c, c). It is found that the OLS estimate for the slope coefficient is not invariant under Lorentz velocity transformation. Accordingly, an alternative estimator for the parameters of linear regression under special relativity is proposed. This estimator can be considered a generalization of the OLS estimator under special relativity; when c approaches to infinity, the proposed estimator and its variance converges to the OLS estimator and its variance, respectively. The variance of the proposed estimator is larger than that of the OLS estimator, which implies that hypothesis testing using the OLS estimator and its variance may result in a liberal test under special relativity.

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