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Likelihood Based Inference and Bias Reduction in the Modified Skew-t-Normal Distribution

Author

Listed:
  • Jaime Arrué

    (Departamento de Estadística y Ciencias de Datos, Facultad de Ciencias Básicas, Universidad de Antofagasta, Antofagasta 1240000, Chile)

  • Reinaldo B. Arellano-Valle

    (Departamento de Estadística, Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago 7820436, Chile)

  • Enrique Calderín-Ojeda

    (Centre for Actuarial Studies, Department of Economics, The University of Melbourne, Melbourne, VIC 3010, Australia)

  • Osvaldo Venegas

    (Departamento de Ciencias Matemáticas y Físicas, Facultad de Ingeniería, Universidad de Católica de Temuco, Temuco 4780000, Chile)

  • Héctor W. Gómez

    (Departamento de Estadística y Ciencias de Datos, Facultad de Ciencias Básicas, Universidad de Antofagasta, Antofagasta 1240000, Chile)

Abstract

In this paper, likelihood-based inference and bias correction based on Firth’s approach are developed in the modified skew-t-normal (MStN) distribution. The latter model exhibits a greater flexibility than the modified skew-normal (MSN) distribution since it is able to model heavily skewed data and thick tails. In addition, the tails are controlled by the shape parameter and the degrees of freedom. We provide the density of this new distribution and present some of its more important properties including a general expression for the moments. The Fisher’s information matrix together with the observed matrix associated with the log-likelihood are also given. Furthermore, the non-singularity of the Fisher’s information matrix for the MStN model is demonstrated when the shape parameter is zero. As the MStN model presents an inferential problem in the shape parameter, Firth’s method for bias reduction was applied for the scalar case and for the location and scale case.

Suggested Citation

  • Jaime Arrué & Reinaldo B. Arellano-Valle & Enrique Calderín-Ojeda & Osvaldo Venegas & Héctor W. Gómez, 2023. "Likelihood Based Inference and Bias Reduction in the Modified Skew-t-Normal Distribution," Mathematics, MDPI, vol. 11(15), pages 1-21, July.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:15:p:3287-:d:1203039
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    References listed on IDEAS

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    1. Arellano-Valle, Reinaldo B. & Ferreira, Clécio S. & Genton, Marc G., 2018. "Scale and shape mixtures of multivariate skew-normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 166(C), pages 98-110.
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