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

The inverse Gaussian (IG) family is strikingly analogous to the Gaussian 2 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 rootbeta(1) and beta(2), routinely employed in statistical analyses. We use known similarities between the two families to define a concept termed IG-symnietry, an analogue of the symmetry, and to develop IG-analogues delta(1) and delta(2) of rootbeta(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 rootb(1) and b(2). Some applications are discussed, and the analogies between the two families, enhanced during this study are tabulated.