Abstract
The inverse Gaussian (IG) family is strikingly analogous to the Gaussian family in terms of having simple inference solutions, which use the familiar χ2, 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 β2, 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 δ1 and δ2 of \(\sqrt {\beta _1 } \) and β2, respectively. Interestingly, the asymptotic null distributions of the sample versions d1, d2 of δ1, δ2 are exactly the same as those of their normal counterparts \(\sqrt {b_1 } \) and b2. Some applications are discussed, and the analogies between the two families, enhanced during this study are tabulated.
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