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Cauchy or not Cauchy? New goodness-of-fit tests for the Cauchy distribution

Author

Listed:
  • Bruno Ebner

    (Karlsruhe Institute of Technology (KIT))

  • Lena Eid

    (Karlsruhe Institute of Technology (KIT))

  • Bernhard Klar

    (Karlsruhe Institute of Technology (KIT))

Abstract

We introduce a new characterization of the Cauchy distribution and propose a class of goodness-of-fit tests for the Cauchy family. The limit distribution is derived in a Hilbert space framework under the null hypothesis. The new tests are consistent against a large class of alternatives. A comparative Monte Carlo simulation study shows that the test is a good competitor for the state of the art procedures, and we apply the tests to log-returns of cryptocurrencies.

Suggested Citation

  • Bruno Ebner & Lena Eid & Bernhard Klar, 2024. "Cauchy or not Cauchy? New goodness-of-fit tests for the Cauchy distribution," Statistical Papers, Springer, vol. 65(1), pages 45-78, February.
    • Handle: RePEc:spr:stpapr:v:65:y:2024:i:1:d:10.1007_s00362-022-01382-0
    DOI: 10.1007/s00362-022-01382-0
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    References listed on IDEAS

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    1. N. Henze, 1990. "An approximation to the limit distribution of the epps-pulley test statistic for normality," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 37(1), pages 7-18, December.
    2. Besbeas, Panagiotis & Morgan, Byron J. T., 2001. "Integrated squared error estimation of Cauchy parameters," Statistics & Probability Letters, Elsevier, vol. 55(4), pages 397-401, December.
    3. 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.
    4. Jin Zhang, 2002. "Powerful goodness‐of‐fit tests based on the likelihood ratio," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(2), pages 281-294, May.
    5. Nora Gürtler & Norbert Henze, 2000. "Goodness-of-Fit Tests for the Cauchy Distribution Based on the Empirical Characteristic Function," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(2), pages 267-286, June.
    6. M. Mahdizadeh & Ehsan Zamanzade, 2017. "New goodness of fit tests for the Cauchy distribution," Journal of Applied Statistics, Taylor & Francis Journals, vol. 44(6), pages 1106-1121, April.
    7. Bora H. Onen & Dennis C. Dietz & Vincent C. Yen & Albert H. Moore, 2001. "Goodness-of-fit tests for the Cauchy distribution," Computational Statistics, Springer, vol. 16(1), pages 97-107, March.
    8. Bernhard Klar & Simos Meintanis, 2012. "Specification tests for the response distribution in generalized linear models," Computational Statistics, Springer, vol. 27(2), pages 251-267, June.
    9. Christian Goldmann & Bernhard Klar & Simos Meintanis, 2015. "Data transformations and goodness-of-fit tests for type-II right censored samples," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 78(1), pages 59-83, January.
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