On the Superposition of Overlapping Realizations of a Poisson Process

• Activity Number: 320
• Type: Contributed
• Date/Time: Tuesday, August 5, 2008 : 2:00 PM to 3:50 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Gustavo L. Gilardoni*+ and Enrico A. Colosimo
• Companies: Universidade de Brasilia and Universidade Federal de Minas Gerais
• Keywords: greatest convex minorant ; nonparametric intensity estimation ; optimal maintenance time ; recurrent events ; repairable systems

Abstract

Many nonparametric procedures are well established when only one realization of a nonhomogeneous Poisson processes (NHPP) is observed. In this note it is argued that the case when several realizations are observed over different time intervals can be reduced to the former one. More precisely, it is shown that, when time is measured in the Total Time on Test scale, the superposition of overlapping realizations of a NHPP is also a NHPP and is sufficient for inferential purposes. Thus, using results already available when only one realization is observed, one can obtain for the many realizations setting, for instance, (i) the maximum likelihood estimator of a monotonic intensity or (ii) bootstrap confidence bands based on Kernel estimates of the intensity. The techniques are illustrated using a real data set containing the failure history of 40 power transformers.