Letac, Gérard; Seshadri, Vanamamalai The expectation of \(X^{-1}\) as a function of \({\mathbb{E}}(X)\) for an exponential family on the positive line. (English) Zbl 0694.62006 Ann. Stat. 17, No. 4, 1735-1741 (1989). 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. Cited in 3 Documents MSC: 62E10 Characterization and structure theory of statistical distributions 60E10 Characteristic functions; other transforms Keywords:reciprocal of a random variable; gamma distributions; inverse-Gaussian distributions; Ressel families; natural exponential family on the positive real line; expectation; affine function; Abel families × Cite Format Result Cite Review PDF Full Text: DOI