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

Author

Listed:
  • Shaul K. Bar-Lev

    (Faculty of Industrial Engineering and Technology Management, Holon Institute of Technology, Holon 6810201, Israel
    These authors contributed equally to this work.)

  • Apostolos Batsidis

    (Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece
    These authors contributed equally to this work.)

  • Jochen Einbeck

    (Department of Mathematical Sciences and Research Methods Centre, Durham University, Durham DH13LE, UK
    These authors contributed equally to this work.)

  • Xu Liu

    (School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai 200433, China
    These authors contributed equally to this work.)

  • Panpan Ren

    (School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai 200433, China)

Abstract

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.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:7:p:1603-:d:1107781
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    References listed on IDEAS

    as
    1. Shaul K. Bar-Lev, 2020. "Independent, Tough Identical Results: The Class of Tweedie on Power Variance Functions and the Class of Bar-Lev and Enis on Reproducible Natural Exponential Families," International Journal of Statistics and Probability, Canadian Center of Science and Education, vol. 9(1), pages 1-30, January.
    2. Célestin C. Kokonendji & Aboubacar Y. Touré & Rahma Abid, 2022. "On General Exponential Weight Functions and Variation Phenomenon," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 84(2), pages 924-940, August.
    3. Shaul K. Bar-Lev & Ad Ridder, 2019. "Monte Carlo Methods for Insurance Risk Computation," International Journal of Statistics and Probability, Canadian Center of Science and Education, vol. 8(3), pages 1-54, November.
    4. Villaseñor, José A. & González-Estrada, Elizabeth, 2015. "A variance ratio test of fit for Gamma distributions," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 281-286.
    5. 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.
    6. Shaul K. Bar-Lev & Muriel Casalis, 2003. "A Classification of Reproducible Natural Exponential Families in the Broad Sense," Journal of Theoretical Probability, Springer, vol. 16(1), pages 175-196, January.
    Full references (including those not matched with items on IDEAS)

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    Cited by:

    1. Shaul K. Bar-Lev, 2023. "The Exponential Dispersion Model Generated by the Landau Distribution—A Comprehensive Review and Further Developments," Mathematics, MDPI, vol. 11(20), pages 1-23, October.

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