Statistical Properties of Inverse Gaussian Distributions. II

Abstract

Given a fixed number $n$ of observations on a variate $x$ which has the Inverse Gaussian probability density function $$\exp\big\{-\frac{\phi^2x}{2\lambda} + \phi - \frac{\lambda}{2x}\big\} \sqrt{\frac{\lambda}{2\pi x^3}},\quad 0 < x < \infty$$, for which $E(x) = \lambda/\phi = \mu$, it is shown how to find functions of the sample mean $m$ whose expectations can be expressed suitably in terms of the parameter $\phi$ (or $\mu$). In particular, it is shown that the conditional expectation of any unbiased estimator $\tilde{\kappa}_r$ of the $r$th cumulant $\kappa_r$ is $$E(\tilde{\kappa}_r \mid m) = 2m(\frac{1}{2}\lambda n^2)^{r - 1}e^{\frac{1}{2}g} \int^\infty_1 (u - 1)^{2r - 3}e^{-\frac{1}{2}gu^2} du/(r - 2)!$$ where $g = \lambda n/m$. This expectation may be evaluated either by series given in the paper or by using published tables of numerical values of certain functions to which it can be related. The conditional variance of the usual mean square estimator $s^2$ of $\kappa_2$ is also found. These results give an asymptotic series for the conditional variance of a generalization $\chi^2_s = (n - 1)s^2/E(s^2 \mid m)$ of a statistic discussed by Cochran. Exact formulae for the expectation of the statistic $s^2/m^3$ and its mean square error as an estimator of $\lambda^{-1}$ are given or described. This statistic is a consistent estimator of $\lambda^{-1}$ and has asymptotically an efficiency of $\phi/(\phi + 3)$.