DISTRIBUTION-FUNCTIONS AND LOG-CONCAVITY

This paper presents a collection of log-concavity results of one-dimensional cumulative distribution functions (cdf's) F(x, theta) and the related functions F(x, theta)BAR = 1 - F(x, theta). J(c)(x, theta) = F(x + c, theta) - F(x, theta), c > 0, in both x is-an-element-of R or x is-an-element-of Z and theta is-an-element-of THETA, where R denotes the real line and Z the set of integers. We give a review of results available in the literature and try to fill some gaps in this field. It is well-known that log-concavity properties in x of a density f carry over to F, FBAR, and J(c) in the continuous and discrete case. In addition, it will be seen that the log-concavity of g(y) = f(e(y)) in y for a Lebesgue density f with f(x) = 0 for x < 0 implies the log-concavity of F. This criterion applies to many common densities. Moreover, a convex statistic T defined on R(n) is shown to have a log-concave cdf whenever the underlying n-dimensional Lebesgue density h is log-concave. A slight generalization of the approach in Das Gupta & Sarkar (1984) is used to establish a connection between log-concavity in x of probability densities f or cdf's F and log-concavity of F, FBAR, and J(c) in theta not only in the real, but also in the discrete case. Finally we apply the theory to the most common univariate distributions and discuss some further results obtained in the literature by different methods.