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For a given parametric probability model, we consider the risk of the maximumlikelihood estimator with respect to $\alpha$-divergence, which includes thespecial cases of Kullback--Leibler divergence, the Hellinger distance and$\chi^2$ divergence. The asymptotic expansion of the risk is given with respectto sample sizes of up to order $n^{-2}$. Each term in the expansion isexpressed with the geometrical properties of the Riemannian manifold formed bythe parametric probability model. We attempt to measure the difficulty ofspecifying a model through this expansion.

Asymptotic expansion of the risk of maximum likelihood estimator with  respect to $\alpha$-divergence as a measure of the difficulty of specifying a  parametric model -- with detailed proof