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.