M-estimation for linear models with exchangeable errors

Abstract

This paper focuses on the estimation of parameters in linear models with fixed regressors and exchangeable errors, especially, these errors are assumed to be a subset of an infinite exchangeable sequence. We propose a modified M-estimation to estimate the slope parameter, which outperforms the traditional M-estimation when the random errors are exchangeable. The modified M-estimation utilizes a design point centralization technique, leading to a more suitable estimator with improved theoretical properties. To establish the asymptotic properties of the modified M-estimator, we also develop a weighted central limit theorem for exchangeable sequences, which is of independent interest. Regarding the intercept, we prove that there is no consistent estimator for the intercept when the exchangeable errors have non-zero covariance, and we propose an unbiased estimator for the intercept as a viable compromise. Based on our theoretical findings, we develop a simulation-based approach to approximate the confidence set of the slope, and we establish a prediction model specifically designed for linear models with exchangeable errors. Additionally, we illustrate the finite sample performance of the proposed method through various numerical simulations and a real data application.