AISM 54, 138-154

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

### Govind S. Mudholkar1 and Rajeshwari Natarajan2

1Department of Statistics, University of Rochester, Rochester, NY 14627, U.S.A.
2Department 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.

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