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

Shaul K. Bar-Lev, Apostolos Batsidis, Jochen Einbeck, Xu Liu, Panpan Ren

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Article number1603
Issue number7
StatePublished - Apr 2023
Externally publishedYes

Bibliographical note

Funding Information:
This research was funded by the project Establishment of capacity building infrastructures in Biomedical Research (BIOMED-20) (MIS 5047236), which is implemented under Action Reinforcement of the Research and Innovation Infrastructure, funded by Operational Programme Competitiveness, Entrepreneurship and Innovation (NSRF 2014-2020) and co-financed by Greece and the European Union (European Regional Development Fund). National Natural Science Foundation of China (12271329).

Publisher Copyright:
© 2023 by the authors.


  • asymptotic distribution
  • bootstrap
  • goodness-of-fit tests
  • Monte Carlo simulation
  • natural exponential family
  • power variance function
  • Tweedie scale

ASJC Scopus subject areas

  • Computer Science (miscellaneous)
  • Mathematics (all)
  • Engineering (miscellaneous)


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