Selection of model selection criteria for multivariate ridge regression

In the present study, we consider the selection of model selection criteria for multivariate ridge regression. There are several model selection criteria for selecting the ridge parameter in multivariate ridge regression, e.g., the C-p criterion and the modified C-p (MCp) criterion. We propose the generalized C-p (GC(p)) criterion, which includes C-p and MCp criteria as special cases. The GC(p) criterion is specified by a non-negative parameter lambda, which is referred to as the penalty parameter. We attempt to select an optimal penalty parameter such that the predicted mean squared error (PMSE) of the predictor of the ridge regression after optimizing the ridge parameter is minimized. Through numerical experiments, we verify that the proposed optimization methods exhibit better performance than conventional optimization methods, i.e., optimizing only the ridge parameter by minimizing the C-p or MCp criterion.