Concentration reversals in ridge regression

Ridge regression is often the method of choice for approaching ill-conditioned systems. A canonical form identifies regions in the parameter space where Ordinary Least Squares (OLS) is problematic. A curious but unrecognized property of ridge solutions emerges: Under spherical errors with or without moments, the relative concentrations of the canonical estimators reverse as the ridge scalar evolves, the estimators least concentrated under OLS being most concentrated under ridge regression, and conversely.