  In this paper, we propose a discrete circular distribution obtained by
extending the wrapped Poisson distribution. This new distribution, the
Invariant Wrapped Poisson (IWP), enjoys numerous advantages: simple tractable
density, parameter-parsimony and interpretability, good circular dependence
structure and easy random number generation thanks to known
marginal/conditional distributions. Existing discrete circular distributions
strongly depend on the initial direction and orientation, i.e. a change of the
reference system on the circle may lead to misleading inferential results. We
investigate the invariance properties, i.e. invariance under change of initial
direction and of the reference system orientation, for several continuous and
discrete distributions. We prove that the introduced IWP distribution satisfies
these two crucial properties. We estimate parameters in a Bayesian framework
and provide all computational details to implement the algorithm. Inferential
issues related to the invariance properties are discussed through numerical
examples on artificial and real data.
