AISM 54, 138-154

© 2002 ISM

## The inverse Gaussian models : Analogues of symmetry, skewness and kurtosis

### Govind S. Mudholkar^{1} and Rajeshwari Natarajan^{2}

^{1}Department of Statistics, University of Rochester,
Rochester, NY 14627, U.S.A.

^{2}Department of Biostatistics,
University of Rochester Medical Center, 601 Elmwood Avenue, Box 630, Rochester, NY 14642,
U.S.A.

(Received July 19, 1999; revised May 8, 2000)

Abstract.
The inverse Gaussian ($IG$) family is strikingly analogous to the Gaussian
family in terms of having simple inference solutions, which use the familiar
$\chi^2$, $t$ and $F$ distributions, for a variety of basic problems. Hence,
the $IG$ family, consisting of asymmetric distributions is widely used for
modelling and analyzing nonnegative skew data. However, the process lacks measures of
model appropriateness corresponding to $\sqrt{\beta_1}$ and $\beta_2$, routinely
employed in statistical analyses. We use known similarities between the two families
to define a concept termed $IG$-symmetry, an analogue of the symmetry, and to
develop $IG$-analogues $\delta_1$ and $\delta_2$ of $\sqrt{\beta_1}$ and $\beta_2$,
respectively. Interestingly, the asymptotic null distributions of the sample
versions $d_1$, $d_2$ of $\delta_1$, $\delta_2$ are exactly the same as those
of their normal counterparts $\sqrt{b_1}$ and $b_2$. Some applications are
discussed, and the analogies between the two families, enhanced during this
study are tabulated.

Key words and phrases:
Contaminated inverse Gaussian distribution, goodness-of-fit tests,
$IG$-scale mixtures.

**Source**
(TeX , DVI )