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An Alternative to the Log-Skew-Normal Distribution: Properties, Inference, and an Application to Air Pollutant Concentrations

Author

Listed:
  • Jaime Arrué

    (Departamento de Matemáticas, Facultad de Ciencias Básicas, Universidad de Antofagasta, Antofagasta 1240000, Chile)

  • Reinaldo Boris Arellano-Valle

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

  • Osvaldo Venegas

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

  • Heleno Bolfarine

    (Departamento de Estatítica, IME, Universidade de São Paulo, São Paulo 05508-090, Brazil)

  • Héctor W. Gómez

    (Departamento de Matemáticas, Facultad de Ciencias Básicas, Universidad de Antofagasta, Antofagasta 1240000, Chile)

Abstract

In this study, we consider an alternative to the log-skew-normal distribution. It is called the modified log-skew-normal distribution and introduces greater flexibility in the skewness and kurtosis parameters. We first study several of the main probabilistic properties of the new distribution, such as the computation of its moments and the non-existence of the moment-generating function. Parameter estimation by the maximum likelihood approach is also studied. This approach presents an overestimation problem in the shape parameter, which in some cases, can even be infinite. However, as we demonstrate, this problem is solved by adapting bias reduction using Firth’s approach. We also show that the modified log-skew-normal model likewise inherits the non-singularity of the Fisher information matrix of the untransformed model, when the shape parameter is null. Finally, we apply the modified log-skew-normal model to a real example related to pollution data.

Suggested Citation

  • Jaime Arrué & Reinaldo Boris Arellano-Valle & Osvaldo Venegas & Heleno Bolfarine & Héctor W. Gómez, 2022. "An Alternative to the Log-Skew-Normal Distribution: Properties, Inference, and an Application to Air Pollutant Concentrations," Mathematics, MDPI, vol. 10(22), pages 1-14, November.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:22:p:4336-:d:977233
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