Log-concavity and log-convexity in passage time densities of diffusion and birth-death processes

Abstract

Diffusion and birth-death processes have basic theoretical and practical importance for statistics. Insight into the structure of transition distributions and passage time distributions for such processes has been given in recent years by Feller. Karlin, Kemperman, D. G. Kendall, Reuter and many others. An elementary account of this work and partial bibliography has been given elsewhere ([9], [10)]. Certain key passage time densities and sojourn time densities for such processes have a simple property of log-concavity or log-convexity and associated unimodality. Such properties provide information on the character of distributions unavailable from the spectral representations, Laplace transforms and series of convolutions at hand. These properties may also have value for purposes of estimation and optimization.