Definition of Log-Concave / Log Concavity: A function f(w) is said to be log-concave if its natural log, ln(f(w)) is a concave function; that is, assuming f is differentiable, f''(w)/f(w) - f'(w)² <= 0 Since log is a strictly concave function, any concave function is also log-concave.

A random variable is said to be log-concave if its density function is log-concave. The uniform, normal, beta, exponential, and extreme value distributions have this property. If pdf f() is log-concave, then so is its cdf F() and 1-F(). The truncated version of a log-concave function is also log-concave. In practice the intuitive meaning of the assumption that a distribution is log-concave is that (a) it doesn't have multiple separate maxima (although it could be flat on top), and (b) the tails of the density function are not "too thick".

An equivalent definition, for vector-valued random variables, is in Heckman and Honore, 1990, p 1127. Random vector X is log-concave iff its density f() satisfies the condition that f(ax₁+(1-a)x₂)≥[f(x₁) ]^a[f(x₂)]^(1-a) for all x₁, and x₂ in the support of X and all a satisfying 0≤a≤1. (Econterms)

Terms related to Log-Concave / Log Concavity:

• Random Variable