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Schur convexity of the maximum likelihood function for the multivariate hypergeometric and multinomial distributions

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  • Boland, Philip J.
  • Proschan, Frank

Abstract

We define for a family distributions p[theta](x), [theta] [epsilon] [Theta], the maximum likelihood function L at a sample point x by L(x) = sup[theta][epsilon][Theta]P[theta](x). We show that for the multivariate hypergeometric and multinomial families, the maximum likelihood function is a Schur convex function of x. In the language of majorization, this implies that the more diverse the elements or components of x are, the larger is the function L(x). Several applications of this result are given in the areas of parameter estimation and combinatorics. An improvement and generalization of a classical inequality of Khintchine is also derived as a consequence.

Suggested Citation

  • Boland, Philip J. & Proschan, Frank, 1987. "Schur convexity of the maximum likelihood function for the multivariate hypergeometric and multinomial distributions," Statistics & Probability Letters, Elsevier, vol. 5(5), pages 317-322, August.
    • Handle: RePEc:eee:stapro:v:5:y:1987:i:5:p:317-322
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    Cited by:

    1. Requena, F. & Ciudad, N. Martin, 2000. "Characterization of maximum probability points in the Multivariate Hypergeometric distribution," Statistics & Probability Letters, Elsevier, vol. 50(1), pages 39-47, October.
    2. Ginés Almagro-Hernández & Juana-María Vivo & Manuel Franco & Jesualdo Tomás Fernández-Breis, 2021. "Analysing the Protein-DNA Binding Sites in Arabidopsis thaliana from ChIP-seq Experiments," Mathematics, MDPI, vol. 9(24), pages 1-26, December.
    3. Chu, Yu-Ming & Xia, Wei-Feng & Zhang, Xiao-Hui, 2012. "The Schur concavity, Schur multiplicative and harmonic convexities of the second dual form of the Hamy symmetric function with applications," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 412-421.

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