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A New Look at the Dirichlet Distribution: Robustness, Clustering, and Both Together

Author

Listed:
  • Salvatore D. Tomarchio

    (University of Catania)

  • Antonio Punzo

    (University of Catania)

  • Johannes T. Ferreira

    (University of Pretoria)

  • Andriette Bekker

    (University of Pretoria
    Centre for Environmental Studies)

Abstract

Compositional data have peculiar characteristics that pose significant challenges to traditional statistical methods and models. Within this framework, we use a convenient mode parametrized Dirichlet distribution across multiple fields of statistics. In particular, we propose finite mixtures of unimodal Dirichlet (UD) distributions for model-based clustering and classification. Then, we introduce the contaminated UD (CUD) distribution, a heavy-tailed generalization of the UD distribution that allows for a more flexible tail behavior in the presence of atypical observations. Thirdly, we propose finite mixtures of CUD distributions to jointly account for the presence of clusters and atypical points in the data. Parameter estimation is carried out by directly maximizing the maximum likelihood or by using an expectation-maximization (EM) algorithm. Two analyses are conducted on simulated data to illustrate the effects of atypical observations on parameter estimation and data classification, and how our proposals address both aspects. Furthermore, two real datasets are investigated and the results obtained via our models are discussed.

Suggested Citation

  • Salvatore D. Tomarchio & Antonio Punzo & Johannes T. Ferreira & Andriette Bekker, 2025. "A New Look at the Dirichlet Distribution: Robustness, Clustering, and Both Together," Journal of Classification, Springer;The Classification Society, vol. 42(1), pages 31-53, March.
    • Handle: RePEc:spr:jclass:v:42:y:2025:i:1:d:10.1007_s00357-024-09480-4
    DOI: 10.1007/s00357-024-09480-4
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    References listed on IDEAS

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    1. Salvatore D. Tomarchio & Antonio Punzo, 2019. "Modelling the loss given default distribution via a family of zero‐and‐one inflated mixture models," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 182(4), pages 1247-1266, October.
    2. Salvatore D. Tomarchio & Antonio Punzo, 2020. "Dichotomous unimodal compound models: application to the distribution of insurance losses," Journal of Applied Statistics, Taylor & Francis Journals, vol. 47(13-15), pages 2328-2353, November.
    3. Bertin, Karine & Genest, Christian & Klutchnikoff, Nicolas & Ouimet, Frédéric, 2023. "Minimax properties of Dirichlet kernel density estimators," Journal of Multivariate Analysis, Elsevier, vol. 195(C).
    4. Tanita Botha & Johannes Ferreira & Andriette Bekker, 2021. "Alternative Dirichlet Priors for Estimating Entropy via a Power Sum Functional," Mathematics, MDPI, vol. 9(13), pages 1-17, June.
    5. Nolan, John P., 1998. "Parameterizations and modes of stable distributions," Statistics & Probability Letters, Elsevier, vol. 38(2), pages 187-195, June.
    6. J. Aitchison & I. J. Lauder, 1985. "Kernel Density Estimation for Compositional Data," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 34(2), pages 129-137, June.
    7. Eva Fiserova & Karel Hron, 2010. "Total least squares solution for compositional data using linear models," Journal of Applied Statistics, Taylor & Francis Journals, vol. 37(7), pages 1137-1152.
    8. Chen, Song Xi, 1999. "Beta kernel estimators for density functions," Computational Statistics & Data Analysis, Elsevier, vol. 31(2), pages 131-145, August.
    9. Lawrence Hubert & Phipps Arabie, 1985. "Comparing partitions," Journal of Classification, Springer;The Classification Society, vol. 2(1), pages 193-218, December.
    10. Luca Bagnato & Antonio Punzo, 2013. "Finite mixtures of unimodal beta and gamma densities and the $$k$$ -bumps algorithm," Computational Statistics, Springer, vol. 28(4), pages 1571-1597, August.
    11. Calif, Rudy & Emilion, Richard & Soubdhan, Ted, 2011. "Classification of wind speed distributions using a mixture of Dirichlet distributions," Renewable Energy, Elsevier, vol. 36(11), pages 3091-3097.
    12. Antonio Punzo, 2019. "A new look at the inverse Gaussian distribution with applications to insurance and economic data," Journal of Applied Statistics, Taylor & Francis Journals, vol. 46(7), pages 1260-1287, May.
    13. Ongaro, A. & Migliorati, S., 2013. "A generalization of the Dirichlet distribution," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 412-426.
    14. Ouimet, Frédéric & Tolosana-Delgado, Raimon, 2022. "Asymptotic properties of Dirichlet kernel density estimators," Journal of Multivariate Analysis, Elsevier, vol. 187(C).
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