  This paper reviews and compares popular methods, some old and some very
recent, that produce time series having Poisson marginal distributions. The
paper begins by narrating ways where time series with Poisson marginal
distributions can be produced. Modeling nonstationary series with covariates
motivates consideration of methods where the Poisson parameter depends on time.
Here, estimation methods are developed for some of the more flexible methods.
The results are used in the analysis of 1) a count sequence of tropical
cyclones occurring in the North Atlantic Basin since 1970, and 2) the number of
no-hitter games pitched in major league baseball since 1893. Tests for whether
the Poisson marginal distribution is appropriate are included.
