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Maximum entropy characterizations of the multivariate Liouville distributions


  • Bhattacharya, Bhaskar


A random vector X=(X1,X2,...,Xn) with positive components has a Liouville distribution with parameter [theta]=([theta]1,[theta]2,...,[theta]n) if its joint probability density function is proportional to , [theta]i>0 [R.D. Gupta, D.S.P. Richards, Multivariate Liouville distributions, J. Multivariate Anal. 23 (1987) 233-256]. Examples include correlated gamma variables, Dirichlet and inverted Dirichlet distributions. We derive appropriate constraints which establish the maximum entropy characterization of the Liouville distributions among all multivariate distributions. Matrix analogs of the Liouville distributions are considered. Some interesting results related to I-projection from a Liouville distribution are presented.

Suggested Citation

  • Bhattacharya, Bhaskar, 2006. "Maximum entropy characterizations of the multivariate Liouville distributions," Journal of Multivariate Analysis, Elsevier, vol. 97(6), pages 1272-1283, July.
  • Handle: RePEc:eee:jmvana:v:97:y:2006:i:6:p:1272-1283

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    References listed on IDEAS

    1. Peddada, Shyamal Das & Richards, Donald St. P., 1991. "Entropy inequalities for some multivariate distributions," Journal of Multivariate Analysis, Elsevier, vol. 39(1), pages 202-208, October.
    2. Silviu Guiasu, 1990. "A classification of the main probability distributions by minimizing the weighted logarithmic measure of deviation," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 42(2), pages 269-279, June.
    3. Zografos, K. & Nadarajah, S., 2005. "Expressions for Rényi and Shannon entropies for multivariate distributions," Statistics & Probability Letters, Elsevier, vol. 71(1), pages 71-84, January.
    4. Gupta, Rameshwar D. & Richards, Donald St.P., 1987. "Multivariate Liouville distributions," Journal of Multivariate Analysis, Elsevier, vol. 23(2), pages 233-256, December.
    5. Karlin, Samuel & Rinott, Yosef, 1980. "Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions," Journal of Multivariate Analysis, Elsevier, vol. 10(4), pages 467-498, December.
    6. Gokhale, D. V., 1983. "On entropy-based goodness-of-fit tests," Computational Statistics & Data Analysis, Elsevier, vol. 1(1), pages 157-165, March.
    7. Zografos, K., 1999. "On Maximum Entropy Characterization of Pearson's Type II and VII Multivariate Distributions," Journal of Multivariate Analysis, Elsevier, vol. 71(1), pages 67-75, October.
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