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Negative Association Does not Imply Log-Concavity of the Rank Sequence

Part of: Distribution theory - Probability Multivariate analysis

Published online by Cambridge University Press:  14 July 2016

Klas Markström
Klas Markström*
Affiliation:
Umeå University
*
∗Postal address: Department of Mathematics and Mathematical Statistics, Umeå University, SE-901 87 Umeå, Sweden. Email address: klas.markstrom@math.umu.se
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Abstract

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We present a minimum counterexample to the conjecture that a negatively associated random variable has an ultra-log-concave rank sequence. The rank sequence does not in fact even need to be unimodal.

Keywords

Negative associationrank sequence

MSC classification

Secondary: 62H20: Measures of association (correlation, canonical correlation, etc.) 60E05: Distributions
Type
Short Communications
Information
Journal of Applied Probability , Volume 44 , Issue 4 , December 2007 , pp. 1119 - 1121
Copyright
Copyright © Applied Probability Trust 2007 

References

[1] Borcea, J., Brändén, P. and Liggett, T. M. (2007). Negative dependence and the geometry of polynomials. Available at http://www.arxiv.org/abs/0707.2340.Google Scholar
[2] Liggett, T. M. (1997). Ultra logconcave sequences and negative dependence. J. Combin. Theory Ser. A 79, 315–325.Google Scholar
[3] Pemantle, R. (2000). Towards a theory of negative dependence. Probabilistic techniques in equilibrium and nonequilibrium statistical physics. J. Math. Phys. 41, 1371–1390.Google Scholar