Bias correction of cross-validation criterion based on Kullback-Leibler information under a general condition

This paper deals with the bias correction of the cross-validation (CV) criterion to estimate the predictive Kullback-Leibler information. A bias-corrected CV criterion is proposed by replacing the ordinary maximum likelihood estimator with the maximizer of the adjusted log-likelihood function. The adjustment is just slight and simple, but the improvement of the bias is remarkable. The bias of the ordinary CV criterion is O(n-1), but that of the bias-corrected CV criterion is O(n-2). We verify that our criterion has smaller bias than the AIC, TIC, EIC and the ordinary CV criterion by numerical experiments.