A Robust Coefficient of Determination Based on Implicit Weighting

Abstract

In the linear regression model, the standard coefficient of determination $R^2$ and its weighted counterpart are commonly used to assess the quality of the linear fit. However, both metrics are susceptible to the influence of outliers and heteroskedasticity within the dataset. This paper introduces a robust version of $R^2$, based on the least weighted squares (LWS) estimator, and examines its statistical properties in detail. We investigate the impact of data quantization on $R^2$ and its robust variants, and propose a hypothesis test for assessing the equality of expected values between two $R^2$ versions. Numerical experiments on 29 publicly available datasets reveal that confidence intervals for the LWS-based coefficient of determination are generally narrower than those for existing measures, especially in homoskedastic settings. In contrast, under heteroskedasticity, narrower intervals do not necessarily imply greater robustness, highlighting the nuanced behavior of these estimators. The comparison with the well-known least trimmed squares (LTS) estimator underscores the promise of the LWS approach, which exhibits favorable efficiency properties and more reliable interval estimation in many practical scenarios.