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We introduce a novel family of projected distributions on the circle and thesphere, namely the circular and spherical projected Cauchy distributions aspromising alternatives for modeling circular and directional data. The circulardistribution encompasses the wrapped Cauchy distribution as a special case,featuring a more convenient parameterisation. Next, we propose a generalisedwrapped Cauchy distribution that includes an extra parameter, enhancing the fitof the distribution. In the spherical context, we impose two conditions on thescatter matrix, resulting in an elliptically symmetric distribution. Ourprojected distributions exhibit attractive properties, such as closed-formnormalising constants and straightforward random value generation. Thedistribution parameters can be estimated using maximum likelihood and we assesstheir bias through numerical studies. We compare our proposed distributions toexisting models with real data sets, demonstrating superior fit both with andwithout covariates.

Circular and Spherical Projected Cauchy Distributions: A Novel Framework for Circular and Directional Data Modeling