A Small-Sample Choice of the Tuning Parameter in Ridge Regression.

We propose new approaches for choosing the shrinkage parameter in ridge regression, a penalized likelihood method for regularizing linear regression coefficients, when the number of observations is small relative to the number of parameters. Existing methods may lead to extreme choices of this parameter, which will either not shrink the coefficients enough or shrink them by too much. Within this "small-, large-" context, we suggest a correction to the common generalized cross-validation (GCV) method that preserves the asymptotic optimality of the original GCV. We also introduce the notion of a "hyperpenalty", which shrinks the shrinkage parameter itself, and make a specific recommendation regarding the choice of hyperpenalty that empirically works well in a broad range of scenarios. A simple algorithm jointly estimates the shrinkage parameter and regression coefficients in the hyperpenalized likelihood. In a comprehensive simulation study of small-sample scenarios, our proposed approaches offer superior prediction over nine other existing methods.