A versatile trivariate wrapped Cauchy copula with applications to toroidal and cylindrical data

In this paper, we propose a new flexible distribution for data on the three-dimensional torus which we call a trivariate wrapped Cauchy copula. Our trivariate copula has several attractive properties. It has a simple form of density and is unimodal. its parameters are interpretable and allow adjustable degree of dependence between every pair of variables and these can be easily estimated. The conditional distributions of the model are well studied bivariate wrapped Cauchy distributions. Furthermore, the distribution can be easily simulated. Parameter estimation via maximum likelihood for the distribution is given and we highlight the simple implementation procedure to obtain these estimates. We compare our model to its competitors for analysing trivariate data and provide some evidence of its advantages. Another interesting feature of this model is that it can be extended to cylindrical copula as we describe this new cylindrical copula and then gives its properties. We illustrate our trivariate wrapped Cauchy copula on data from protein bioinformatics of conformational angles, and our cylindrical copula using climate data related to buoy in the Adriatic Sea. The paper is motivated by these real trivariate datasets, but we indicate how the model can be extended to multivariate copulas.