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An entropy inequality for the bi-multivariate hypergeometric distribution. (English) Zbl 0673.62042

A bi-multivariate discrete hypergeometric distribution is the distribution on non-negative integer \(m\times n\) matrices \(A=(a_{ij})\) with row sums \(r=(r_ 1,...,r_ m)\) and column sums \(c=(c_ 1,...,c_ n)\) and with the probability function \[ P(A)=\prod r_ j!\prod c_ j!/(N!\prod a_{ij}!),\quad N=r_ 1+...+r_ m=c_ 1+...+c_ n. \] This can be looked upon as a generalization of the standard multivariate discrete hypergeometric distribution arising from sampling without replacement from a set of N objects consisting of k distinct sets of objects.
The authors show that the entropy of this bi-multivariate hypergeometric distributon is a Schur-concave function of the block-size parameters. The result is proved by elementary combinatorial arguments and using some basic properties of logarithms.
Reviewer: A.Mathai

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
60E15 Inequalities; stochastic orderings
94A17 Measures of information, entropy
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References:

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[2] P. S. Matveev, The entropy of the multinomial distribution, Teor. Verojatnost. i Primenen. 23 (1978), no. 1, 196 – 198 (Russian, with English summary). · Zbl 0388.60015
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