We introduce a novel family of projected distributions on the circle and the sphere, namely the circular and spherical projected Cauchy distributions, as promising alternatives for modelling circular and spherical data. The circular distribution encompasses the wrapped Cauchy distribution as a special case, while featuring a more convenient parameterisation. We also propose a generalised wrapped Cauchy distribution that includes an extra parameter, enhancing the fit of the distribution. In the spherical context, we impose two conditions on the scatter matrix of the Cauchy distribution, resulting in an elliptically symmetric distribution. Our projected distributions exhibit attractive properties, such as a closed-form normalising constant and straightforward random value generation. The distribution parameters can be estimated using maximum likelihood, and we assess their bias through numerical studies. Further, we compare our proposed distributions to existing models with real datasets, demonstrating equal or superior fitting both with and without covariates.

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