Nonhomogeneous Poisson processes and logconcavity

In this article, we identify conditions under which the epoch times and the inter-epoch intervals of a nonhomogeneous Poisson process have logconcave densities. The results are extended to relevation counting processes, We also study discrete-time counting processes and find conditions under which the epoch times and the inter-epoch intervals of these discrete-time processes have logconcave discrete probability densities. The results are interpreted in terms of minimal repair and record values. Several examples illustrate the theory.