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Generalized Linear Model (GLM) Applications for the Exponential Dispersion Model Generated by the Landau Distribution

Author

Listed:
  • Shaul K. Bar-Lev

    (Faculty of Industrial Engineering and Technology Management, Holon Institute of Technology, Holon 5810201, Israel)

  • Xu Liu

    (School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai 200433, China
    Yunnan Key Laboratory of Statistical Modeling and Data Analysis, Yunnan University, Kunming 650091, China)

  • Ad Ridder

    (School of Business and Economics, Vrije University of Amsterdam, 1081 HV Amsterdam, The Netherlands)

  • Ziyu Xiang

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

Abstract

The exponential dispersion model (EDM) generated by the Landau distribution, denoted by EDM-EVF (exponential variance function), belongs to the Tweedie scale with power infinity. Its density function does not have an explicit form and, as of yet, has not been used for statistical aspects. Out of all EDMs belonging to the Tweedie scale, only two EDMs are steep and supported on the whole real line: the normal EDM with constant variance function and the EDM-EVF. All other absolutely continuous steep EDMs in the Tweedie scale are supported on the positive real line. This paper aims to accomplish an overall picture of all generalized linear model (GLM) applications belonging to the Tweedie scale by including the EDM-EVF. This paper introduces all GLM ingredients needed for its analysis, including the respective link function and total and scaled deviance. We study its analysis of deviance, derive the asymptotic properties of the maximum likelihood estimation (MLE) of the covariate parameters, and obtain the asymptotic distribution of deviance, using saddlepoint approximation. We provide numerical studies, which include estimation algorithm, simulation studies, and applications to three real datasets, and demonstrate that GLM using the EDM-EVF performs better than the linear model based on the normal EDM. An R package accompanies all of these.

Suggested Citation

  • Shaul K. Bar-Lev & Xu Liu & Ad Ridder & Ziyu Xiang, 2024. "Generalized Linear Model (GLM) Applications for the Exponential Dispersion Model Generated by the Landau Distribution," Mathematics, MDPI, vol. 12(13), pages 1-21, June.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:13:p:2021-:d:1425352
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    References listed on IDEAS

    as
    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.
    2. Ricci, Lila & Martinez, Raul, 2008. "Adjusted R2-type measures for Tweedie models," Computational Statistics & Data Analysis, Elsevier, vol. 52(3), pages 1650-1660, January.
    3. Philip Hougaard, 1995. "Nonlinear regression and curved exponential families. Improvement of the approximation to the asymptotic distribution," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 42(1), pages 191-202, December.
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