pp 1–28 | Cite as

A new characterization of the Gamma distribution and associated goodness-of-fit tests

  • Steffen Betsch
  • Bruno EbnerEmail author


We propose a class of weighted \(L^2\)-type tests of fit to the Gamma distribution. Our novel procedure is based on a fixed point property of a new transformation connected to a Steinian characterization of the family of Gamma distributions. We derive the weak limits of the statistic under the null hypothesis and under contiguous alternatives. The result on the limit null distribution is used to prove the asymptotic validity of the parametric bootstrap that is implemented to run the tests. Further, we establish the global consistency of our tests in this bootstrap setting, and conduct a Monte Carlo simulation study to show the competitiveness to existing test procedures.


Bootstrap procedure Contiguous alternatives Density approach Gamma distribution Goodness-of-fit tests Stein’s method 



The authors thank Norbert Henze for fruitful discussions and helpful comments on the presentation of the material. They also want to express their gratitude to an anonymous referee, an associate editor, and the journal editor, for their insights during the revision process, which led to a major improvement of the article.

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of StochasticsKarlsruhe Institute of TechnologyKarlsruheGermany

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