Convergence of estimative density: criterion for model complexity and sample size

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.