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On the relationship between connections and the asymptotic properties of predictive distributions
Corcuera Valverde, José Manuel; Giummolè, Federica
Universitat de Barcelona
In a recent paper, Komaki studied the second-order asymptotic properties of predictive distributions, using the Kullback-Leibler divergence as a loss function. He showed 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 a loss function any f divergence. A relationship arises between alpha connections and optimal predictive distributions. In particular, using an alpha divergence to measure the goodness of a predictive distribution, the optimal shift of the estimate distribution is related to alpha-covariant derivatives. The expression that 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.
Geometria diferencial
Connexions (Matemàtica)
Estadística matemàtica
Teoria de la predicció
Differential geometry
Prediction theory
Connections (Mathematics)
Mathematical statistics
(c) ISI/BS, International Statistical Institute, Bernoulli Society, 1999
Bernoulli Society for Mathematical Statistics and Probability

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