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Second order Bayes estimators, being the main tool in second order optimal statistical theory, provide a natural basis for a new approach to the problem of the prediction of functions of expectation functional for members of an exponential dispersion family. A general formula, providing such prediction up to the term of the order 1/n, is suggested and the application to the problem of the prediction of the tail of distributions is demonstrated. The results are illustrated with normal and gammadoi:10.2143/ast.34.2.505144 fatcat:5fdok6t3hjauhhtyrk5fqqcvrm