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Multivariate Liouville distributions

Author

Listed:
  • Gupta, Rameshwar D.
  • Richards, Donald St.P.

Abstract

A random vector (X1, ..., Xn), with positive components, has a Liouville distribution if its joint probability density function is of the formf(x1 + ... + xn)x1a1.1 ... xnan.1 with theai all positive. Examples of these are the Dirichlet and inverted Dirichlet distributions. In this paper, a comprehensive treatment of the Liouville distributions is provided. The results pertain to stochastic representations, transformation properties, complete neutrality, marginal and conditional distributions, regression functions, and total positivity and reverse rule properties. Further, these topics are utilized in various characterizations of the Dirichlet and inverted Dirichlet distributions. Matrix analogs of the Liouville distributions are also treated, and many of the results obtained in the vector setting are extended appropriately.

Suggested Citation

  • Gupta, Rameshwar D. & Richards, Donald St.P., 1987. "Multivariate Liouville distributions," Journal of Multivariate Analysis, Elsevier, vol. 23(2), pages 233-256, December.
    • Handle: RePEc:eee:jmvana:v:23:y:1987:i:2:p:233-256
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    Citations

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    Cited by:

    1. Edward Hoyle & Levent Ali Menguturk, 2020. "Generalised Liouville Processes and their Properties," Papers 2003.11312, arXiv.org, revised May 2020.
    2. Arnold, Barry C. & Villasenor, Jose A., 2013. "On orthogonality of (X+Y) and X/(X+Y) rather than independence," Statistics & Probability Letters, Elsevier, vol. 83(2), pages 584-587.
    3. Bhattacharya, Bhaskar, 2006. "Maximum entropy characterizations of the multivariate Liouville distributions," Journal of Multivariate Analysis, Elsevier, vol. 97(6), pages 1272-1283, July.
    4. Tian, Guo-Liang & Fang, Hong-Bin & Tan, Ming & Qin, Hong & Tang, Man-Lai, 2009. "Uniform distributions in a class of convex polyhedrons with applications to drug combination studies," Journal of Multivariate Analysis, Elsevier, vol. 100(8), pages 1854-1865, September.
    5. Malini Iyengar & Dipak Dey, 2002. "A semiparametric model for compositional data analysis in presence of covariates on the simplex," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 11(2), pages 303-315, December.
    6. Fang, B. Q., 2003. "The skew elliptical distributions and their quadratic forms," Journal of Multivariate Analysis, Elsevier, vol. 87(2), pages 298-314, November.
    7. Jones, M.C. & Marchand, Éric, 2019. "Multivariate discrete distributions via sums and shares," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 83-93.
    8. 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.
    9. 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.
    10. Bhattacharya, P. K. & Burman, Prabir, 1998. "Semiparametric Estimation in the Multivariate Liouville Model," Journal of Multivariate Analysis, Elsevier, vol. 65(1), pages 1-18, April.
    11. Nawaf Mohammed & Edward Furman & Jianxi Su, 2021. "Can a regulatory risk measure induce profit-maximizing risk capital allocations? The case of Conditional Tail Expectation," Papers 2102.05003, arXiv.org, revised Aug 2021.
    12. Elena Hadjicosta & Donald Richards, 2020. "Integral transform methods in goodness-of-fit testing, II: the Wishart distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(6), pages 1317-1370, December.
    13. Thomas, Seemon & Jacob, Joy, 2006. "A generalized Dirichlet model," Statistics & Probability Letters, Elsevier, vol. 76(16), pages 1761-1767, October.
    14. Denuit, Michel & Robert, Christian Y., 2020. "Conditional tail expectation decomposition and conditional mean risk sharing for dependent and conditionally independent risks," LIDAM Discussion Papers ISBA 2020018, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    15. Volkmar Henschel, 2002. "Statistical inference in simplicially contoured sample distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 56(3), pages 215-228, December.
    16. Belzile, Léo R. & Nešlehová, Johanna G., 2017. "Extremal attractors of Liouville copulas," Journal of Multivariate Analysis, Elsevier, vol. 160(C), pages 68-92.
    17. Kamps, Udo & Rauwolf, Diana, 2023. "A record-values property of a renewal process with random inspection time," Statistics & Probability Letters, Elsevier, vol. 195(C).
    18. McNeil, Alexander J. & Neslehová, Johanna, 2010. "From Archimedean to Liouville copulas," Journal of Multivariate Analysis, Elsevier, vol. 101(8), pages 1772-1790, September.
    19. Ng, Kai Wang & Tang, Man-Lai & Tan, Ming & Tian, Guo-Liang, 2008. "Grouped Dirichlet distribution: A new tool for incomplete categorical data analysis," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 490-509, March.
    20. Michel Denuit & Christian Y. Robert, 2022. "Conditional Tail Expectation Decomposition and Conditional Mean Risk Sharing for Dependent and Conditionally Independent Losses," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1953-1985, September.
    21. Ongaro, A. & Migliorati, S., 2013. "A generalization of the Dirichlet distribution," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 412-426.
    22. Mohammed, Nawaf & Furman, Edward & Su, Jianxi, 2021. "Can a regulatory risk measure induce profit-maximizing risk capital allocations? The case of conditional tail expectation," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 425-436.

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