ABSTRACT
Log-concavity is an important part of information economics. Since the logarithm of the cumulative distribution function (c.d.f.) of a random variable is a concave function, it turns out that the ratio of the probability density function (p.d.f.) to c.d.f. is a monotone decreasing function. Every concave function that is nonnegative on its domain is log-concave. However, the converse is not necessarily true. Properties of log-concave functions are as follows: sum, product integral and convolution. The product of log-concave functions is also log-concave. The Laplace transform of a nonnegative convex function is log-concave. If two independent random variables have log-concave p.d.f.s, then their sum has a log-concave p.d.f. The following probability distributions are log-concave: normal distribution and multivariate normal distributions; exponential distribution; uniform distribution over any convex set; logistic distribution; extreme value distribution; Laplace distribution; chi-distribution; Wishart distribution; Dirichlet distribution; Gamma distribution; beta distribution and Weibull distribution.
