Linear Regression under Special Relativity

TL;DR

This paper examines how ordinary least squares regression behaves under special relativity constraints, revealing non-invariance of estimates and proposing a generalized estimator suited for relativistic contexts.

Contribution

It introduces a new estimator for linear regression parameters under special relativity, generalizing OLS and addressing its limitations in relativistic settings.

Findings

OLS estimates are not Lorentz invariant.

The proposed estimator converges to OLS as c approaches infinity.

Hypothesis tests using OLS may be liberal under relativistic conditions.

Abstract

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.