The trivariate wrapped Cauchy copula – a multi-purpose model for angular data

In this paper, we will present a new flexible distribution for three-dimensional angular data, or data on the three-dimensional torus. Our trivariate wrapped Cauchy copula has the following benefits: (i) simple form of density, (ii) adjustable degree of dependence between every pair of variables, (iii) interpretable and well-estimable parameters, (iv) well-known conditional distributions, (v) a simple data generating mechanism, (vi) unimodality. Moreover, our construction allows for linear marginals, implying that our copula can also model cylindrical data. Parameter estimation via maximum likelihood is explained, a comparison with the competitors in the existing literature is given, and two real datasets are considered, one concerning protein dihedral angles and another about data obtained by a buoy in the Adriatic Sea.