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In this paper, we propose a discrete circular distribution obtained byextending the wrapped Poisson distribution. This new distribution, theInvariant Wrapped Poisson (IWP), enjoys numerous advantages: simple tractabledensity, parameter-parsimony and interpretability, good circular dependencestructure and easy random number generation thanks to knownmarginal/conditional distributions. Existing discrete circular distributionsstrongly depend on the initial direction and orientation, i.e. a change of thereference system on the circle may lead to misleading inferential results. Weinvestigate the invariance properties, i.e. invariance under change of initialdirection and of the reference system orientation, for several continuous anddiscrete distributions. We prove that the introduced IWP distribution satisfiesthese two crucial properties. We estimate parameters in a Bayesian frameworkand provide all computational details to implement the algorithm. Inferentialissues related to the invariance properties are discussed through numericalexamples on artificial and real data.

On initial direction, orientation and discreteness in the analysis of circular variables