• Bernoulli
  • Volume 5, Number 1 (1999), 163-176.

On the relationship between α connections and the asymptotic properties of predictive distributions

José M. Corcuera and Federica Giummolè

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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 α connections and optimal predictive distributions. In particular, using an α divergence to measure the goodness of a predictive distribution, the optimal shift of the estimate distribution is related to α-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.

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Bernoulli, Volume 5, Number 1 (1999), 163-176.

First available in Project Euclid: 12 March 2007

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curved exponential family differential geometry f divergences predictive distributions second-order asymptotic theory α connections α embedding curvature


Corcuera, José M.; Giummolè, Federica. On the relationship between α connections and the asymptotic properties of predictive distributions. Bernoulli 5 (1999), no. 1, 163--176.

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