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Maximum likelihood estimation of triangular and polygonal distributions

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  • Nguyen, Hien D.
  • McLachlan, Geoffrey J.

Abstract

Triangular distributions are a well-known class of distributions that are often used as elementary example of a probability model. In the past, enumeration and order statistics-based methods have been suggested for the maximum likelihood (ML) estimation of such distributions. A novel parametrization of triangular distributions is presented. The parametrization allows for the construction of an MM (minorization–maximization) algorithm for the ML estimation of triangular distributions. The algorithm is shown to both monotonically increase the likelihood evaluations, and be globally convergent. Using the parametrization is then applied to construct an MM algorithm for the ML estimation of polygonal distributions. This algorithm is shown to have the same numerical properties as that of the triangular distribution. Numerical simulations are provided to demonstrate the performances of the new algorithms against established enumeration and order statistics-based methods.

Suggested Citation

  • Nguyen, Hien D. & McLachlan, Geoffrey J., 2016. "Maximum likelihood estimation of triangular and polygonal distributions," Computational Statistics & Data Analysis, Elsevier, vol. 102(C), pages 23-36.
    • Handle: RePEc:eee:csdana:v:102:y:2016:i:c:p:23-36
    DOI: 10.1016/j.csda.2016.04.003
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    References listed on IDEAS

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    1. Nguyen, Hien D. & McLachlan, Geoffrey J., 2016. "Laplace mixture of linear experts," Computational Statistics & Data Analysis, Elsevier, vol. 93(C), pages 177-191.
    2. Zhou, Hua & Zhang, Yiwen, 2012. "EM vs MM: A case study," Computational Statistics & Data Analysis, Elsevier, vol. 56(12), pages 3909-3920.
    3. Hien Nguyen & Geoffrey McLachlan, 2015. "Maximum likelihood estimation of Gaussian mixture models without matrix operations," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 9(4), pages 371-394, December.
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    5. Nguyen, Hien D. & McLachlan, Geoffrey J. & Wood, Ian A., 2016. "Mixtures of spatial spline regressions for clustering and classification," Computational Statistics & Data Analysis, Elsevier, vol. 93(C), pages 76-85.
    6. Small, Kenneth A. & Ng, Chen Feng, 2014. "Optimizing road capacity and type," Economics of Transportation, Elsevier, vol. 3(2), pages 145-157.
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