Estimation of Poisson Arrival Processes Under Linear Models

In this paper, we consider the problem of estimating the parameters of a Poisson arrival process, where the intensity function is assumed to lie in the span of a known basis. Our goal is to estimate the basis expansions coefficients given a realization of this process. We establish novel guarantees concerning the accuracy achieved by the maximum likelihood estimate. Our initial result is near-optimal, with the exception of an undesirable dependence on the dynamic range of the intensity function. We then show how to remove this dependence through a process of “noise regularization,” which results in an improved bound under our analysis. We conjecture that a similar guarantee should be possible when using a more direct (deterministic) regularization scheme. We conclude with a discussion of practical applications and an empirical examination of the proposed regularization schemes.