Theoretical Study of Some Angle Parameter Trigonometric Copulas

Abstract

Copulas are important probabilistic tools to model and interpret the correlations of measures involved in real or experimental phenomena. The versatility of these phenomena implies the need for diverse copulas. In this article, we describe and investigate theoretically new two-dimensional copulas based on trigonometric functions modulated by a tuning angle parameter. The independence copula is, thus, extended in an original manner. Conceptually, the proposed trigonometric copulas are ideal for modeling correlations into periodic, circular, or seasonal phenomena. We examine their qualities, such as various symmetry properties, quadrant dependence properties, possible Archimedean nature, copula ordering, tail dependences, diverse correlations (medial, Spearman, and Kendall), and two-dimensional distribution generation. The proposed copulas are fleshed out in terms of data generation and inference. The theoretical findings are supplemented by some graphical and numerical work. The main results are proved using two-dimensional inequality techniques that can be used for other copula purposes.