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Asymptotic expansion of the risk of maximum likelihood estimator with respect to α-divergence

ABSTRACT For a given parametric probability model, we consider the risk of the maximum likelihood estimator with respect to α-divergence, which includes the special cases of Kullback–Leibler divergence, the Hellinger distance, and essentially χ2-divergence. The asymptotic expansion of the risk is given with respect to sample sizes 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.