  We propose a new framework for the modelling of count data exhibiting zero
inflation (ZI). The main part of this framework includes a new and more general
parameterisation for ZI models which naturally includes both over- and
under-inflation. It further sheds new theoretical light on modelling and
inference and permits a simpler alternative, which we term as multiplicative,
in contrast to the dominant mixture and hurdle models. Our approach gives the
statistician access to new types of ZI of which mixture and hurdle are special
cases. We outline a simple parameterised modelling approach which can help to
infer both ZI type and degree and provide an underlying treatment that shows
that current ZI models are themselves typically within the exponential family,
thus permitting much simpler theory, computation and classical inference. We
outline some possibilities for a natural Bayesian framework for inference; and
a rich basis for work on correlated ZI counts.
  The present paper is an incomplete report on the underlying theory. A later
version will include computational issues and provide further examples.
