On the relationships between \alpha-connections and the asymptotic properties of predictive distributions

Abstract

In a recent paper Komaki studies the second-order asymptotic properties of the predictive distributions, using the Kullback-Leibler divergence as loss function. He shows that estimative distributions with asymptotically efficient estimators can be improved by predictive distributions that do not belong to the model. The model is assumed to be a multidimensional curved exponential family. In this paper we generalize the result assuming as loss function any f-divergence. It appears a relationship between the a-connections and the optimal predictive distributions. In particular, using an a-divergence to measure the goodness of a predictive distribution, the optimal shift of the estimative distribution is related with alpha-covariant derivatives. The expression we obtain for the asymptotic risk is also useful to study the higher-order asymptotic properties of an estimator, in the mentioned class of loss functions.