The Use and Interpretation of Residuals Based on Robust Estimation

Abstract

Residual plots and diagnostic techniques are important tools for examining the fit of a regression model. In the case of least squares fits, plots of residuals provide a visual assessment of the adequacy of various aspects of the fitted model. An important question is whether plots of robust residuals can be interpreted in the same manner as their least squares counterparts. This article addresses this problem for two popular classes of robust estimates: M estimates and GM estimates of the Mallows and Schweppe types. First-order properties of the residuals and fitted values are derived under correct and misspecified models. These properties are insightful on the general interpretability of robust residual plots and on their ability to detect curvature in misspecified models. The results of a simulation study consisting of tests for randomness and curvature in residual plots supports these theoretical properties. Standardization of robust residuals is also presented.

Key Words:

• GM estimate
• High breakdown
• Linear model
• M estimate
• Regression diagnostic
• Residual plot