AISM 54, 138-154
© 2002 ISM
(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.