  Fern\'andez-Dur\'an and Gregorio-Dom\'inguez (2014) defined a family of
probability distributions for a vector of circular random variables by
considering multiple nonnegative trigonometric sums. These distributions are
highly flexible and can present numerous modes and skewness. Several operations
on these multivariate distributions were translated into operations on the
vector of parameters; for instance, marginalization involves calculating the
eigenvectors and eigenvalues of a matrix, and independence among subsets of the
vector of circular variables translates to a Kronecker product of the
corresponding subsets of the vector of parameters. Furthermore, it was
demonstrated that the family of multivariate circular distributions based on
nonnegative trigonometric sums is closed under marginalization and
conditioning, that is, the marginal and conditional densities of any order are
also members of the family. The derivation of marginal and conditional
densities from the joint multivariate density is important when applying this
model in practice to real datasets. A goodness-of-fit test based on the
characteristic function and an alternative parameter estimation algorithm for
high-dimensional circular data was presented and applied to a real dataset on
the daily times of occurrence of maxima and minima of prices in financial
markets.
