Introduction to robust model selection in linear regression

Mallows’ $C_p$ is widely used for the selection of a model from among many non-nested regression models. However, the statistic is a function of two residual sums of squares; it is an aggregate statistic, a function of all the observations. Thus $C_p$ suffers from the well-known lack of robustness of least squares and provides no evidence of whether or how individual observations or unidentified structure are affecting the choice of model. In this part of the toolbox we use the robustness of the data-driven flexible trimming provided by the forward search to choose regression models in the presence of outliers. Our tools are new distributional results on added t-test (Atkinson and Riani, 2002) and $C_p$ (Riani and Atkinson, 2010) in the forward search and a powerful new version of the $C_p$ plot, which we call a generalized candlestick plot.