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

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  • Steffen Betsch

    (Karlsruhe Institute of Technology)

  • Bruno Ebner

    (Karlsruhe Institute of Technology)

Abstract

We propose a class of weighted $$L^2$$ 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.

Suggested Citation

  • Steffen Betsch & Bruno Ebner, 2019. "A new characterization of the Gamma distribution and associated goodness-of-fit tests," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(7), pages 779-806, October.
    • Handle: RePEc:spr:metrik:v:82:y:2019:i:7:d:10.1007_s00184-019-00708-7
    DOI: 10.1007/s00184-019-00708-7
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    References listed on IDEAS

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    5. L. Baringhaus & B. Ebner & N. Henze, 2017. "The limit distribution of weighted $$L^2$$ L 2 -goodness-of-fit statistics under fixed alternatives, with applications," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(5), pages 969-995, October.
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    7. Chen, Xiaohong & White, Halbert, 1998. "Central Limit And Functional Central Limit Theorems For Hilbert-Valued Dependent Heterogeneous Arrays With Applications," Econometric Theory, Cambridge University Press, vol. 14(2), pages 260-284, April.
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    Cited by:

    1. Steffen Betsch & Bruno Ebner, 2021. "Fixed point characterizations of continuous univariate probability distributions and their applications," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(1), pages 31-59, February.
    2. E. Bothma & J. S. Allison & I. J. H. Visagie, 2022. "New classes of tests for the Weibull distribution using Stein’s method in the presence of random right censoring," Computational Statistics, Springer, vol. 37(4), pages 1751-1770, September.
    3. Shaul K. Bar-Lev & Apostolos Batsidis & Jochen Einbeck & Xu Liu & Panpan Ren, 2023. "Cumulant-Based Goodness-of-Fit Tests for the Tweedie, Bar-Lev and Enis Class of Distributions," Mathematics, MDPI, vol. 11(7), pages 1-20, March.

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