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Fern\'andez-Dur\'an and Gregorio-Dom\'inguez (2014) defined a family ofprobability distributions for a vector of circular random variables byconsidering multiple nonnegative trigonometric sums. These distributions arehighly flexible and can present numerous modes and skewness. Several operationson these multivariate distributions were translated into operations on thevector of parameters; for instance, marginalization involves calculating theeigenvectors and eigenvalues of a matrix, and independence among subsets of thevector of circular variables translates to a Kronecker product of thecorresponding subsets of the vector of parameters. Furthermore, it wasdemonstrated that the family of multivariate circular distributions based onnonnegative trigonometric sums is closed under marginalization andconditioning, that is, the marginal and conditional densities of any order arealso members of the family. The derivation of marginal and conditionaldensities from the joint multivariate density is important when applying thismodel in practice to real datasets. A goodness-of-fit test based on thecharacteristic function and an alternative parameter estimation algorithm forhigh-dimensional circular data was presented and applied to a real dataset onthe daily times of occurrence of maxima and minima of prices in financialmarkets.

Multivariate Nonnegative Trigonometric Sums Distributions for High-Dimensional Multivariate Circular Data