For the multivariate linear regression model with unknown covariance, the corrected Akaike information criterion is the minimum variance unbiased estimator of the expected Kullback--Leibler discrepancy. In this study, based on the loss estimation framework, we show its inadmissibility as an estimator of the Kullback--Leibler discrepancy itself, instead of the expected Kullback--Leibler discrepancy. We provide improved estimators of the Kullback--Leibler discrepancy that work well in reduced-rank situations and examine their performance numerically.

翻译：针对协方差未知的多元线性回归模型，修正后的Akaike信息准则是对期望Kullback--Leibler差异的最小方差无偏估计。本研究基于损失估计框架，证明了该准则作为Kullback--Leibler差异本身（而非期望Kullback--Leibler差异）估计量的不可容许性。我们提出了在降秩场景下性能优异的改进型Kullback--Leibler差异估计量，并通过数值实验检验了其表现。