On New Types of Multivariate Trigonometric Copulas

Abstract

Copulas are useful functions for modeling multivariate distributions through their univariate marginal distributions and dependence structures. They have a wide range of applications in all fields of science that deal with multivariate data. While there is a plethora of copulas, those based on trigonometric functions, especially in dimensions greater than two, have received much less attention. They are, however, of interest because of the properties of oscillation and periodicity of the trigonometric functions, which can appear in certain models of correlation of natural phenomena. In order to fill this gap, this paper introduces and investigates two new types of “multivariate trigonometric copulas”. Their main theoretical properties are studied, and some perspectives for applications are sketched for future work. In particular, we show that the proposed copulas are symmetric, not associative, with no orthant dependence, and with copula densities that have wide oscillations, which remains an uncommon property in the field. The expressions of their multivariate Spearman’s rho are also determined. Furthermore, the first type of the proposed copulas has the interesting feature of having a multivariate Spearman’s rho equal to 0 for all of the dimensions. Some graphic evidence supports the findings. Some mathematical formulas involving the product of n trigonometric functions may be of independent interest.

Theoretical Advancements on a Few New Dependence Models Based on Copulas with an Original Ratio Form

“…For the purpose of this paper, the copula defined in Equation ( 6) is called the ratio-sine (RS) copula. It belongs to the family of trigonometric copulas, which have gained a lot of attention these last few years (see, for instance, [10,12,13,28]). They can be used to uncover dependencies hidden in variables based on circular data (see [28]).…”

“…As a result, many authors have devised novel strategies for producing copulas with unique forms and manageable dependence properties. See [10][11][12][13][14][15][16][17][18][19][20][21], among others. In particular, the contemporary works of [14,17] have attracted our attention.…”

Theoretical Contributions to Three Generalized Versions of the Celebioglu–Cuadras Copula

“…Moreover, with the tremendous developments in computer calculation, the subject of the copula is more interesting than ever for modern theoretical and practical problems. Recent contributions on this topic can be found in [9][10][11][12][13][14][15][16].…”

Theoretical Study of Some Angle Parameter Trigonometric Copulas

“…In particular, they are ideal for analyzing correlations involved in movement data, circular data, and environmental data. The theory of the classical trigonometric copulas can be found in [5][6][7][8][9][10][11]. For practice, we refer to [12][13][14][15][16].…”

“…where f (x) is a simple one-dimensional function involving a trigonometric function, and θ is the angle parameter that only modulates this trigonometric function. Despite the potential for modeling correlations into periodic, circular, or seasonal phenomena, this area of research appears to have received little attention; none of the references [5][6][7][8][9][10][11] consider such an angle parameter approach. Thus, we describe the most intuitive of such angle parameter copulas, and study them on the theoretical side with a maximum of details.…”