Testing for monotonicity in the intensity of a nonhomogeneous Poisson process

Based on the times of occurrences of a nonhomogeneous Poisson process in a fixed time interval, a pseudo-likelihood ratio test of the null hypothesis that the intensity is nondecreasing with the alternative that it is not nondecreasing is developed. The modifications needed to test for a nonincreasing intensity are also presented. For these statistics, the collection of constant intensities is shown to be least favorable within the collection of nondecreasing intensities. For constant intensities, a suitable approximation to the conditional distribution of the likelihood ratio statistics, given the number of occurrences, is obtained and the approximate conditional tests are studied. Through Monte Carlo work certain types of alternatives are identified which require a large number of occurrences for the likelihood ratio test to detect with reasonable probability. Some inferences about nonhomogeneous Poisson processes are based on a researcher's a priori belief that the intensity is nondecreasing (nonincreasing). These tests provide a statistical method for assessing the validity of the researcher's belief.Likelihood ratio tests Maximum likelihood estimates Order-restricted inferences Repairable systems Trends