Nonhomogeneous Poisson processes as overhaul models

Hollander and Proschan (1974) studied nonhomogeneous Poisson processes as models for systems subject to overhauls. They did not postulate a functional form for the intensity, but showed that certain basic assumptions about the deterioration of the system implied that the mean function is superadditive. They studied tests of the null hypothesis that the intensity is constant with the alternative restricted to superadditive mean functions. For estimation purposes, the class of superadditive mean functions is too broad. We assume that the intensity is nondecreasing between overhauls and that at an overhaul it does not fall below its average prior to the overhaul. These two assumptions imply that the mean function is star-shaped. We obtain the restricted maximum-likelihood estimates under these two assumptions and under the star-shaped restriction. The two estimates are compared on a data set.