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For a parametric model of distributions, the closest distribution in themodel 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 "informationprojection." The estimation risk of the maximum likelihood estimator (MLE) isdefined as the expectation of K-L divergence between the information projectionand the predictive distribution with plugged-in MLE. Here, the asymptoticexpansion of the risk is derived up to $n^{-2}$-order, and the sufficientcondition on the risk for the Bayes error rate between the true distributionand the information projection to be lower than a specified value isinvestigated. Combining these results, the "$p-n$ criterion" is proposed, whichdetermines whether the MLE is sufficiently close to the information projectionfor the given model and sample. In particular, the criterion for an exponentialfamily model is relatively simple and can be used for a complex model with noexplicit form of normalizing constant. This criterion can constitute a solutionto the sample size or model acceptance problem. Use of the $p-n$ criteria isdemonstrated for two practical datasets. The relationship between the resultsand information criteria is also studied.

MLE convergence speed to information projection of exponential family: Criterion for model dimension and sample size -- complete proof version--