  Most of the stationary first-order autoregressive integer-valued (INAR(1))
models were developed for a given thinning operator using either the forward
approach or the backward approach. In the forward approach the marginal
distribution of the time series is specified and an appropriate distribution
for the innovation sequence is sought. Whereas in the backward setting, the
roles are reversed. The common distribution of the innovation sequence is
specified and the distributional properties of the marginal distribution of the
time series are studied. In this article we focus on the backward approach in
presence of the Binomial thinning operator. We establish a number of
theoretical results which we proceed to use to develop stationary INAR(1)
models with finite mean. We illustrate our results by presenting some new
INAR(1) models that show underdispersion.
