  In this paper we consider the problem of estimating the parameters of a
Poisson arrival process where the rate 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 rate function. We then show how to remove this dependence through
a process of "noise regularization", which results in an improved bound. 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.
