A Simulation Study of a Change-Point Poisson Process Based on Two Well-known Test Statistics

Abstract

Methods for analysis of trends for the series of events have been extensively studied by many authors, both from a parametric and a nonparametric point of view. Most of the work on the development of the procedures for the nonhomoge- neous Poisson processes are based on the fixed sample size tests. It is also desirable that in some situations, interim analyses are undertaken periodically. Suppose, for example, a repairable system is under development. A development program might consist of testing to identify deficiencies, a redesign effort to correct the deficiencies, and further testing to verify these corrections and identify new problem areas. It would be advantageous to track the reliability growth trend of the system, by means of the failure data collected during development testing, so that the program could be revised, if necessary, in order to attain the system reliability objectives. Since the data often occur naturally in a sequential fashion, it will be useful to have sequential procedures allowing for repeated significance tests to the accumulating data. In this paper, we provide side-to-side information for the following two widely used test statistics in this area: (1) the well-known Laplace test (called L test), and (2) the most powerful test for the shape parameter in the Poisson process with Weibull intensity (called Z test). Discussion of major results based on a Monte Carlo simulation study include: (1) the estimated probability of type I error at or before the nth test in sampling from a simple Poisson distribution at a constant nominal level, (2) the existing support from the well-developed sequential clinical trial designs available in the literature, (3) performance assessment for abrupt changes (increasing or decreasing in the intensity of the process), and (4) a control charting procedure which presents a visual interpretation of the trend and can be practically translated for tabular or manual use in modeling the occurrences of stochastic phenomena.