A framework for analyzing the periodically-observed time-homogeneous Poisson process

When counting process data are collected from real-world systems, the arrival of each event is often reported in periodic time units (e.g., hour, day, week, month) and the precise arrival time is lost. This periodic reporting introduces discretization error into arrival time data, fundamentally changing the resulting interarrival distribution and inhibiting comparisons to continuous-time stochastic processes (e.g., Poisson process). This article formulates the periodically-observed time-homogeneous Poisson process (PTPP) to account for discretization due to periodic observation when the underlying system is a time-homogeneous Poisson process. In contrast with the analogous Poisson process, the PTPP is not a renewal process; however, its arrivals can be modeled by an infinite-state discrete-time Markov chain with two state variables: the recorded interarrival time and order of the event within the current observation period. The marginal limiting distribution for the first variable (i.e., the limiting interarrival distribution) is derived along with its cumulative distribution, moment generating function, first two moments, and variance. This article shows, through a simulation-based experiment, that there exist a range of discretization-levels for which neither the interarrival nor counting distribution can effectively identify a periodically-observed Poisson process through goodness-of-fit testing; the PTPP model bridges this gap.