  We propose a novel class of count time series models alternative to the
classic Galton-Watson process with immigration (GWI) and Bernoulli offspring. A
new offspring mechanism is developed and its properties are explored. This
novel mechanism, called geometric thinning operator, is used to define a class
of modified GWI (MGWI) processes, which induces a certain non-linearity to the
models. We show that this non-linearity can produce better results in terms of
prediction when compared to the linear case commonly considered in the
literature. We explore both stationary and non-stationary versions of our MGWI
processes. Inference on the model parameters is addressed and the finite-sample
behavior of the estimators investigated through Monte Carlo simulations. Two
real data sets are analyzed to illustrate the stationary and non-stationary
cases and the gain of the non-linearity induced for our method over the
existing linear methods. A generalization of the geometric thinning operator
and an associated MGWI process are also proposed and motivated for dealing with
zero-inflated or zero-deflated count time series data.
