Parameter Estimation and Hypotheses Testing for Nonhomogeneous Poisson Process: Part 2. Numerical Examples

We consider data processing for a Nonhomogeneous Poisson process (NHPP) with Log-linear and Power-law intensity functions. Parameter estimation is carried out by the maximum likelihood method. For the case of the known intensity function, testing the hypothesis that the given sample path is a realization of NHPP, can be accomplished using the fact, that under the NHPP model the mean value functions of NHPP, computed in sequence of ordered failure times, are the failure times of Homogeneous Poisson Process (HPP) with constant intensity function of one, and the intervals between events in the HPP form a sample of i.i.d. standard exponential random variables. Thus it is possible to use standard goodness-of fit tests to check the exponentiality of the process. The computer-intensive procedure for testing the hypothesis that the given sample path belongs to NHPP without making the assumption that the intensity function is known was described in our previous article (Frenkel at al. (2003)). In this article we describe the different goodness-of fit tests and demonstrate our method on the failure data which exist in literature and for our own failure data for the Schlosser Vibration Machine. We also demonstrate the several methods for generating families of stochastic processes with the known probabilistic structure, which includes both NHPP and not NHPP, and check how our method recognizes the underlying process. These processes were used for testing of power properties of goodness-of-fit tests.