## The Annals of Statistics

### The Expectation of $X^1$ as a Function of $\mathbb{E}(X)$ for an Exponential Family on the Positive Line

#### Abstract

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.

#### Article information

Source
Ann. Statist., Volume 17, Number 4 (1989), 1735-1741.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176347391

Digital Object Identifier
doi:10.1214/aos/1176347391

Mathematical Reviews number (MathSciNet)
MR1026309

Zentralblatt MATH identifier
0694.62006

JSTOR
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. 17 (1989), no. 4, 1735--1741. doi:10.1214/aos/1176347391. https://projecteuclid.org/euclid.aos/1176347391