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.
Publié le : 1989-12-14
Classification:
Natural exponential families,
reciprocal of a random variable,
gamma distributions,
inverse-Gaussian distributions,
Ressel families,
Abel families,
62E10,
60E10
@article{1176347391,
author = {Letac, Gerard and Seshadri, Vanamamalai},
title = {The Expectation of $X^1$ as a Function of $\mathbb{E}(X)$ for an Exponential Family on the Positive Line},
journal = {Ann. Statist.},
volume = {17},
number = {1},
year = {1989},
pages = { 1735-1741},
language = {en},
url = {http://dml.mathdoc.fr/item/1176347391}
}
Letac, Gerard; Seshadri, Vanamamalai. The Expectation of $X^1$ as a Function of $\mathbb{E}(X)$ for an Exponential Family on the Positive Line. Ann. Statist., Tome 17 (1989) no. 1, pp. 1735-1741. http://gdmltest.u-ga.fr/item/1176347391/