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The History of the Dirichlet and Liouville Distributions

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  • Rameshwar D. Gupta
  • Donald St. P. Richards

Abstract

The Dirichlet distributions have long been the subject of intense scrutiny in statistics and probability. Despite the enormous interest in, and wide‐ranging applications of, these distributions, little appears to be known about their history. In this article we review the development of the Dirichlet distributions and their companions, the Liouville distributions. After reviewing some integral formulas of Dirichlet and Liouville, we survey the theory and applications of these distributions in statistics. Les distributions de Dirichlet font depuis longtemps l'objet d'études approfondies en statistique et en probabilité. En dépit de l'énorme intéêt portéà ces distributions et de leurs applications variées, on sait peu de choses de leur histoire. Dans se article, on examine le développement des distributions de Dirichlet, ainsi que des distributions associées de liouville. Aprés avoir rappelé quelques formules d'intégrales de Dirichlet et Lioville, on brosse un panorama de la théorie et des applications de ces distributions en statistique.

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  • Rameshwar D. Gupta & Donald St. P. Richards, 2001. "The History of the Dirichlet and Liouville Distributions," International Statistical Review, International Statistical Institute, vol. 69(3), pages 433-446, December.
    • Handle: RePEc:bla:istatr:v:69:y:2001:i:3:p:433-446
    DOI: 10.1111/j.1751-5823.2001.tb00468.x
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    Cited by:

    1. Tremblay, Raymond L. & Tyre, Andrew J. & Pérez, Maria-Eglée & Ackerman, James D., 2021. "Population projections from holey matrices: Using prior information to estimate rare transition events," Ecological Modelling, Elsevier, vol. 447(C).
    2. Yizhen Wang & Menglei Cui & Jiong Guo & Han Zhang & Yingjie Wu & Fu Li, 2023. "Decay Branch Ratio Sampling Method with Dirichlet Distribution," Energies, MDPI, vol. 16(4), pages 1-17, February.
    3. Hernández-Veleros, Zeus Salvador, 2010. "Heterogeneous growth cycles/Ciclos de crecimiento heterogéneo," Estudios de Economia Aplicada, Estudios de Economia Aplicada, vol. 28, pages 625-650, Diciembre.
    4. Ongaro, A. & Migliorati, S., 2013. "A generalization of the Dirichlet distribution," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 412-426.
    5. Gupta, Rameshwar D. & Richards, Donald St. P., 2002. "Moment Properties of the Multivariate Dirichlet Distributions," Journal of Multivariate Analysis, Elsevier, vol. 82(1), pages 240-262, July.
    6. Tian, Guo-Liang & Tang, Man-Lai & Yuen, Kam Chuen & Ng, Kai Wang, 2010. "Further properties and new applications of the nested Dirichlet distribution," Computational Statistics & Data Analysis, Elsevier, vol. 54(2), pages 394-405, February.
    7. Hoyle, Edward & Hughston, Lane P. & Macrina, Andrea, 2011. "Lévy random bridges and the modelling of financial information," Stochastic Processes and their Applications, Elsevier, vol. 121(4), pages 856-884, April.
    8. Konstancja Bobecka & Jacek Wesołowski, 2007. "The Dirichlet Distribution and Process through Neutralities," Journal of Theoretical Probability, Springer, vol. 20(2), pages 295-308, June.

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