Nonhomogeneous Poisson Processes and Logconcavity

Abstract

In this article, we identify conditions under which the epoch times and the interepoch 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 interepoch 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.

Multivariate dependence of spacings of generalized order statistics

“…2.16). In the particular setting of generalized order statistics, this extends a result given in [23] for univariate marginal distributions of inter-epoch times of a relevation counting process.…”

“…Moreover, if λ F is log-concave, then f (and F ) is also log-concave (cf. [23]). Thus, every η j is log-concave on (α(F ), ω(F )) by assumption.…”

Coherent systems based on sequential order statistics

“…, n − 1 by using this argument, because the density of R i+1 (X * i:n ) is not easily accessible for checking logconcavity. Similar problems concerning logconcavity are discussed in Pellerey et al [30] for nonhomogeneous Poisson processes and relevation counting processes as well as in Cramer [9] and in Chen et al [8] for the GOS.…”

Logconcavity and unimodality of progressively censored order statistics

“…For instance, this holds for the standard exponential distribution (see Theorem 2.5). Pellerey et al (2000) discuss the same problem in terms of occurrence times of nonhomogeneous Poisson processes which can be viewed as record values (cf. Gupta and Kirmani, 1988).…”