  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.
