Moment-Based Goodness-of-Fit Tests for the Inverse Gaussian Distribution

Abstract

Data in statistical practice often consist of nonnegative measurements that exhibit positive skewness. The inverse Gaussian (IG) family of distributions provides a versatile and flexible model for analyzing such data. The most appealing feature of an IG model relates to the many similarities that it shares with the ubiquitous Gaussian (G) model; for example, the optimal procedure for testing homogeneity of several means involves an F-distribution analogous to the F-statistic associated with ANOVA. Mudholkar and Natarajan, in their investigation of G–IG analogies, presented two functions (δ1, δ2) of the moments, termed the coefficients of IG skewness and IG kurtosis. They also discussed the parallelism of these functions with the classical coefficients of skewness and kurtosis, which have extensive goodness-of-fit applications. In this article we consider the sample versions of these functions for constructing goodnessof-fit tests for the IG models. The test statistics are proposed, and their asymptotic normal and chi-squared distributions are derived. Monte Carlo simulations are used to refine the asymptotic distributions and to study the operating characteristics of the tests. We demonstrate that the approach proposed in the article offers an advantage over the empirical distribution function tests, enabling one to address 10 distinct goodness-of-fit problems involving IG models against a variety of restricted IG-skewness and IG-kurtosis alternatives.