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

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