For a parametric model of distributions, the closest distribution in the model to the true distribution located outside the model is considered. Measuring the closeness between two distributions with the Kullback–Leibler divergence, the closest distribution is called the “information projection.” The estimation risk of the maximum likelihood estimator is defined as the expectation of Kullback–Leibler divergence between the information projection and the maximum likelihood estimative density (the predictive distribution with the plugged-in maximum likelihood estimator). Here, the asymptotic expansion of the risk is derived up to the second order in the sample size, and the sufficient condition on the risk for the Bayes error rate between the predictive distribution and the information projection to be lower than a specified value is investigated. Combining these results, the “p/n criterion” is proposed, which determines whether the estimative density is sufficiently close to the information projection for the given model and sample. This criterion can constitute a solution to the sample size or model selection problem. The use of the p/n criteria is demonstrated for two practical datasets.

对于分布的参数模型，考虑模型中最接近位于模型外部的真实分布的分布。用 Kullback-Leibler 散度测量两个分布之间的接近程度，最接近的分布称为“信息投影”。最大似然估计的估计风险被定义为信息投影和最大似然估计密度（插入最大似然估计的预测分布）之间的 Kullback-Leibler 散度的期望。在此，将风险的渐近扩展推导到样本量的二阶，并研究预测分布和信息投影之间的贝叶斯错误率低于指定值的风险的充分条件。提出了“ p / n准则”，该准则确定了估计密度是否足够接近给定模型和样本的信息投影。该标准可以构成样本量或模型选择问题的解决方案。对两个实际数据集演示了p / n标准的使用。