Cumulant-Based Goodness-of-Fit Tests for the Tweedie, Bar-Lev and Enis Class of Distributions

The class of natural exponential families (NEFs) of distributions having power variance functions (NEF-PVFs) is huge (uncountable), with enormous applications in various fields. Based on a characterization property that holds for the cumulants of the members of this class, we developed a novel goodness-of-fit (gof) test for testing whether a given random sample fits fixed members of this class. We derived the asymptotic null distribution of the test statistic and developed an appropriate bootstrap scheme. As the content of the paper is mainly theoretical, we exemplify its applicability to only a few elements of the NEF-PVF class, specifically, the gamma and modified Bessel-type NEFs. A Monte Carlo study was executed for examining the performance of both-the asymptotic test and the bootstrap counterpart-in controlling the type I error rate and evaluating their power performance in the special case of gamma, while real data examples demonstrate the applicability of the gof test to the modified Bessel distribution.

Automated summary

The class of natural exponential families (NEFs) with power variance functions (NEF-PVFs) is extensive and has various applications. In this study, a novel goodness-of-fit (gof) test was developed for fitting random samples to fixed members of this class, based on a characterization property of their cumulants. The asymptotic null distribution of the test statistic and an appropriate bootstrap scheme were derived. The performance of the test and its bootstrap counterpart was evaluated using a Monte Carlo study for the gamma distribution and real data examples for the modified Bessel distribution.