Log-concavity and strong log-concavity: A review

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 strongly log-concavity on ℝ under convolution follows from a fundamental monotonicity result of Efron (1969). 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.

Unrealistic Expectations and Misguided Learning

“…PROOF: As the distribution of ε is log concave, it has finite variance σ 2 = E[|ε t | 2 ] (see, for example, Proposition 5.2 in Saumard and Wellner (2014)). PROOF: As the distribution of ε is log concave, it has finite variance σ 2 = E[|ε t | 2 ] (see, for example, Proposition 5.2 in Saumard and Wellner (2014)).…”

Distinguishing Log-Concavity from Heavy Tails

“…To this end, let X (1) < X (2) be the order statistics. According to subexponential theory (see, in particular, [25]), X (1) , X (2) are asymptotically independent given X 1 + X 2 > d, with X (1) having asymptotic distribution F and X (2) being of the form d + e(d)E with e(d), and E as in the proof of Proposition 1. For large d, this gives…”

“…Log-concavity is a widely studied topic in its own right [1,2]. There also exists substantial literature regarding its connections to probability theory and statistics [3,4].…”

Log-location-scale-log-concave distributions for survival and reliability analysis

“…Distributions with log-concave densities have a number of interesting properties ( [1], [2], [5], [16], Section 4.B, [22], [26]). However, focus here is on the resulting properties of g, obtained after exponential transformation.…”

“…This is because log-concave distributions have light-to-moderate tails. In fact, the heaviest possible tails of f are simple exponential ( [1], [2], [22]), in the sense that f (x) ∼ exp(ξx) for some ξ > 0 as x → −∞ and/or f (x) ∼ exp(−ηx) for some η > 0 as x → ∞ (examples include Laplace and logistic distributions). It is easy to see, however, that the tailweight-increasing property of the exponential transformation allows g to have a heavy tail: for example, if f has an exponential right-tail then g(y) ∼ y −(ηλ+1) as y → ∞, has a power, or Pareto, tail with tail index ηλ.…”