## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 28, Number 2 (1957), 362-377.

### Statistical Properties of Inverse Gaussian Distributions. I

#### Abstract

A report is presented on some statistical properties of the family of probability density functions $$\exp \lbrack -\lambda(x - \mu)^2/2\mu^2x\rbrack\lbrack\lambda/2\pi x^3\rbrack^{1/2}$$ for a variate $x$ and parameters $\mu$ and $\lambda$, with $x, \mu, \lambda$ each confined to $(0, \infty)$. The expectation of $x$ is $\mu$, while $\lambda$ is a measure of relative precision. The chief result is that the ml estimators of $\mu$ and $\lambda$ have stochastically independent distributions, and are of a nature which permits of the construction of an analogue of the analysis of variance for nested classifications. The ml estimator of $\mu$ is the sample mean, and for a fixed sample size $n$ its distribution is of the same family as $x$, with the same $\mu$ but with $\lambda$ replaced by $\lambda n$. The distribution of the ml estimator of the reciprocal of $\lambda$ is of the chi-square type. The probability distribution of $1/x$, and the estimation of certain functions of the parameters in heterogeneous data, are also considered.

#### Article information

**Source**

Ann. Math. Statist., Volume 28, Number 2 (1957), 362-377.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177706964

**Digital Object Identifier**

doi:10.1214/aoms/1177706964

**Mathematical Reviews number (MathSciNet)**

MR110132

**Zentralblatt MATH identifier**

0086.35202

**JSTOR**

links.jstor.org

#### Citation

Tweedie, M. C. K. Statistical Properties of Inverse Gaussian Distributions. I. Ann. Math. Statist. 28 (1957), no. 2, 362--377. doi:10.1214/aoms/1177706964. https://projecteuclid.org/euclid.aoms/1177706964

#### See also

- Part II: M. C. K. Tweedie. Statistical Properties of Inverse Gaussian Distributions. II. Ann. Math. Statist., Volume 28, Number 3 (1957), 696--705.Project Euclid: euclid.aoms/1177706881