  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
(K-L) divergence, the closest distribution is called the "information
projection." The estimation risk of the maximum likelihood estimator (MLE) is
defined as the expectation of K-L divergence between the information projection
and the predictive distribution with plugged-in MLE. Here, the asymptotic
expansion of the risk is derived up to $n^{-2}$-order, and the sufficient
condition on the risk for the Bayes error rate between the true 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 MLE is sufficiently close to the information projection
for the given model and sample. In particular, the criterion for an exponential
family model is relatively simple and can be used for a complex model with no
explicit form of normalizing constant. This criterion can constitute a solution
to the sample size or model acceptance problem. Use of the $p-n$ criteria is
demonstrated for two practical datasets. The relationship between the results
and information criteria is also studied.
