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The expectation of \(X^{-1}\) as a function of \({\mathbb{E}}(X)\) for an exponential family on the positive line. (English) Zbl 0694.62006

Summary: If the distribution of X belongs to a natural exponential family on the positive real line, this note studies the expectation of the reciprocal of X as a function of the expectation m of X and characterizes the cases where this function is an affine function of \(m^{-1}\) as gamma, inverse-Gaussian, Ressel or Abel families.

MSC:

62E10 Characterization and structure theory of statistical distributions
60E10 Characteristic functions; other transforms
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