A note on model selection for small sample regression

The risk estimator called “Direct Eigenvalue Estimator” (DEE) is studied. DEE was developed for small sample regression. In contrast to many existing model selection criteria, derivation of DEE requires neither any asymptotic assumption nor any prior knowledge about the noise variance and the noise distribution. It was reported that DEE performed well in small sample cases but DEE performed a little worse than the state-of-the-art ADJ. This seems somewhat counter-intuitive because DEE was developed for specifically regression problem by exploiting available information exhaustively, while ADJ was developed for general setting. In this paper, we point out that the derivation of DEE includes an inappropriate part, notwithstanding the resultant form of DEE being valid in a sense. As its result, DEE cannot derive its potential. We introduce a class of ‘valid’ risk estimators based on the idea of DEE and show that better risk estimators (mDEE) can be found in the class. By numerical experiments, we verify that mDEE often performs better than or at least equally the original DEE and ADJ.