Some results on tests for Poisson processes

Abstract

SUMMARY It is shown that a test proposed by Barnard for Poisson processes, using certain distributionfree statistics, is not consistent against renewal alternatives. However, empirical evidence is given which suggests that a modification of this test due to Durbin results in relatively powerful tests of the Poisson hypothesis. A simple test based on Durbin's modification is described, and its asymptotic relative efficiency with respect to the asymptotically most powerful test against the alternative of a renewal process with gamma-distributed intervals is given. A central problem in the statistical analysis of stationary series of events (point processes) is to test whether an observed series is a realization of a Poisson process. Denote by A the rate parameter of the Poisson process. The test is for a composite null hypothesis, A being a nuisance parameter. In a Poisson process the numbers of events in non-overlapping intervals are independent Poisson variates, and moreover the intervals between events are independent random variables with the simple exponential distribution. Consequently tests of the Poisson hypothesis can be devised for various alternative hypotheses. Epstein (1960) has surveyed these tests. We will be concerned here with tests which are useful against rather vaguely specified alternatives. There are many such tests and we will try to assess the relative utility of the most important ones. The main source of these tests is an idea of Barnard's (1953). Assume first of all that observation of the series is for a fixed time period (0, t) and that n events occur at times t1, t2..., t., measured from the origin. The idea is to test, conditionally on the number of events in (0, t), N, being equal to n, that the variables Ui = Tilt (i = 1, 2,..., n), are independent variates with distribution 0 (u < 0),