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

TL;DR

This paper introduces a flexible family of multivariate circular distributions based on nonnegative trigonometric sums, with properties like closure under marginalization and conditioning, and demonstrates their practical application in financial data analysis.

Contribution

The paper extends the family of multivariate circular distributions with new properties, a goodness-of-fit test, and a parameter estimation algorithm for high-dimensional data.

Findings

Distributions are closed under marginalization and conditioning.

A goodness-of-fit test based on the characteristic function was developed.

An application to financial market data demonstrated practical utility.

Abstract

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.