Forward and backward tests for an abrupt change in the intensity of a poisson process

Abstract

We show that the Z test for a Poisson process defined in the papers of Bain et al. (1985), and Engelhardt et al. (1990) has an asymmetric performance in detecting an alternative which is a step-function intensity. The Z test has much less power in differentiating between a constant intensity and increasing step- function intensities than between the constant and the decreasing step-function intensities for failure- truncated data. However, it has more power in differentiating between a constant intensity and increasing step-intensities, than between the constant and the decreasing step-intensities for the time-truncated data. Consequently, the results of both studies (Bain et al., 1985, and Engelhardt et al., 1990) which are based on non-decreasing trend alternatives for the time-truncated sampling should not be extended symmetrically to tests with non-increasing trend alternatives for either time-truncated data or failure- truncated data. (In fairness to Bain et al. and Engelhardt et al., they ever recommended its use with decreasing intensities.) In order to maximize the power, we propose that both the forward and the backward tests defined in this article be performed for abrupt changes in equipment performance following scheduled overhauls of a repairable system and, by implication, other applications.

Keywords:

• Laplace test
• Poisson processes
• Step-intensity
• Weibull processes