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2014 Log-concavity and strong log-concavity: A review
Adrien Saumard, Jon A. Wellner
Statist. Surv. 8: 45-114 (2014). DOI: 10.1214/14-SS107

Abstract

We review and formulate results concerning log-concavity and strong-log-concavity in both discrete and continuous settings. We show how preservation of log-concavity and strong log-concavity on $\mathbb{R}$ under convolution follows from a fundamental monotonicity result of Efron (1965). We provide a new proof of Efron’s theorem using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013). Along the way we review connections between log-concavity and other areas of mathematics and statistics, including concentration of measure, log-Sobolev inequalities, convex geometry, MCMC algorithms, Laplace approximations, and machine learning.

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Adrien Saumard. Jon A. Wellner. "Log-concavity and strong log-concavity: A review." Statist. Surv. 8 45 - 114, 2014. https://doi.org/10.1214/14-SS107

Information

Published: 2014
First available in Project Euclid: 9 December 2014

zbMATH: 1360.62055
MathSciNet: MR3290441
Digital Object Identifier: 10.1214/14-SS107

Subjects:
Primary: 60E15, 62E10
Secondary: 62H05

Rights: Copyright © 2014 The author, under a Creative Commons Attribution License

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Vol.8 • 2014
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