Tests for positive jumps in the intensity of a Poisson process: a power study

Tests are considered for the hypothesis of a constant intensity against the alternative of an intensity which increases with time in a nonhomogeneous Poisson process. Attention is focused on step-function alternatives and tests designed for such alternatives. One application is testing for abrupt changes in equipment following scheduled overhauls. The authors recommend the order-restricted likelihood-ratio test over an ordered chi-square test for such situations, provided the points at which jumps can occur are known. Otherwise, they recommend the test based on the Laplace statistic. The performance of these tests is evaluated for smooth alternatives, with the result that the smallest relative power of the order-restricted likelihood-ratio test is 73%, while for the Laplace test it is 82%. A numerical example based on failure times for a main-propulsion diesel engine is presented. The result is that the order-restricted likelihood-ratio test corresponds to the lowest statistical significance level