Statistics Surveys
- Statist. Surv.
- Volume 8 (2014), 45-114.
Log-concavity and strong log-concavity: A review
Adrien Saumard and Jon A. Wellner
Full-text: Open access
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.
Article information
Source
Statist. Surv., Volume 8 (2014), 45-114.
Dates
First available in Project Euclid: 9 December 2014
Permanent link to this document
https://projecteuclid.org/euclid.ssu/1418134163
Digital Object Identifier
doi:10.1214/14-SS107
Mathematical Reviews number (MathSciNet)
MR3290441
Zentralblatt MATH identifier
1360.62055
Subjects
Primary: 60E15: Inequalities; stochastic orderings 62E10: Characterization and structure theory
Secondary: 62H05: Characterization and structure theory
Keywords
Concave convex convolution inequalities log-concave monotone preservation strong log-concave
Citation
Saumard, Adrien; Wellner, Jon A. Log-concavity and strong log-concavity: A review. Statist. Surv. 8 (2014), 45--114. doi:10.1214/14-SS107. https://projecteuclid.org/euclid.ssu/1418134163
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