  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 in spite that the resultant form of DEE is 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.
