Statistics Surveys

Log-concavity and strong log-concavity: A review

Adrien Saumard and Jon A. Wellner

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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.

Article information

Statist. Surv., Volume 8 (2014), 45-114.

First available in Project Euclid: 9 December 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings 62E10: Characterization and structure theory
Secondary: 62H05: Characterization and structure theory

Concave convex convolution inequalities log-concave monotone preservation strong log-concave


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

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