Bivariate Copulas Based on Counter-Monotonic Shock Method

2023

Analytics

Copulas are probabilistic functions that are being used more and more frequently to describe, examine, and model the interdependence of continuous random variables. Among the numerous proposed copulas, renewed interest has recently been shown in the so-called Celebioglu–Cuadras copula. It is mainly because of its simplicity, exploitable dependence properties, and potential for applicability. In this article, we contribute to the development of this copula by proposing three generalized versions of it, each involving three tuning parameters. The main results are theoretical: they consist of determining wide and manageable intervals of admissible values for the involved parameters. The proofs are mainly based on limit, differentiation, and factorization techniques as well as mathematical inequalities. Some of the configuration parameters are new in the literature, and original phenomena are revealed. Subsequently, the basic properties of the proposed copulas are studied, such as symmetry, quadrant dependence, various expansions, concordance ordering, tail dependences, medial correlation, and Spearman correlation. Detailed examples, numerical tables, and graphics are used to support the theory.

Section: Introductionmentioning

confidence: 99%

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

2023

Analytics

“…We may mention the Gaussian copulas, Student copulas, extreme-value copulas, Farlie-Gumbel-Morgenstern (FGM) copulas, and Archimedean copulas. In addition, among the recent refrences, we may mention the new bivariate FGM copulas studied in [6] and [7], the bivariate copulas constructed via a unit Weibull distribution distortion in [8], the bivariate transformation of copulas proposed in [9], the bivariate copulas constructed via the Rüschendorf method in [10], the bivariate copulas based on the counter-monotonic shock method created in [11], the bivariate trigonometric-and hyperbolic-FGM-type copulas developed in [12], the bivariare exponential-type copulas developed in [13], the bivariate circular-linear copulas proposed in [14] and [15], and the bivariate ratio-type copulas elaborated in [16] and [17]. Recent applications of bivariate copulas in applied sectors include [18], [19] and [20].…”

Section: Introductionmentioning

confidence: 99%

Extensions of Two Bivariate Strict Archimedean Copulas

Computational Journal of Mathematical and Statistical Sciences

2023

The copula approach provides an option for capturing the structure of dependence between two quantitative variables. This approach is based on special bivariate functions called copulas. In brief, a copula can be presented as a "functional print" of a structure of dependence. Among the numerous types of copulas, the strict Archimedean copulas are the most famous. However, some of them appear to be unfairly neglected and understudied. We attempt to rehabilitate two of them, namely the Nelsen strict Archimedean copulas numbers 10 and 17 by extending their mathematical functionality. For each of them, the extension is made by the thorough addition of a parameter, which reveals to have a significant impact on several important aspects. As the main results, we determine the range of admissible values for the involved parameters. Then, the basic properties of the proposed extended copulas are studied, including symmetry, tail dependence, bounds, and correlation. A focus is put on the tau of Kendall as the main correlation measure. For one of the extended copulas, it is shown that there are some connections with the Frank copula and that the associated tau of Kendall almost reached the perfect interval of [−1, 1], demonstrating the significance of our findings. Graphics, short statistical works, and numerical tables are given to support the theoretical results.

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

2023

Modelling

Copulas are well-known tools for describing the relationship between two or more quantitative variables. They have recently received a lot of attention, owing to the variable dependence complexity that appears in heterogeneous modern problems. In this paper, we offer five new copulas based on a common original ratio form. All of them are defined with a single tuning parameter, and all reduce to the independence copula when this parameter is equal to zero. Wide admissible domains for this parameter are established, and the mathematical developments primarily rely on non-trivial limits, two-dimensional differentiations, suitable factorizations, and mathematical inequalities. The corresponding functions and characteristics of the proposed copulas are looked at in some important details. In particular, as common features, it is shown that they are diagonally symmetric, but not Archimedean, not radially symmetric, and without tail dependence. The theory is illustrated with numerical tables and graphics. A final part discusses the multi-dimensional variation of our original ratio form. The contributions are primarily theoretical, but they provide the framework for cutting-edge dependence models that have potential applications across a wide range of fields. Some established two-dimensional inequalities may be of interest beyond the purposes of this paper.