composite1COP: Composition of a Single Symmetric Copula with Two Compositing Parameters

Description

The composition of a single copula (Salvadori et al., 2006, p. 266, prop. C.3) is created by the following result related to “composition of copulas” in that reference. Suppose $\mathbf{C}(u,v)$ is a symmetric copula (see COP) with parameters $\Theta$ and $\mathbf{C} \ne \mathbf{\Pi}$ (for $\mathbf{\Pi}$ see P), then a family of generally asymmetric copulas $\mathbf{C}_{\alpha,\beta; \Theta}$ with two compositing parameters $0 < \alpha,\beta < 1$, and $\alpha \ne \beta$, which also includes just the copula $\mathbf{C}(u,v)$ as a limiting case for $\alpha = \beta = 0$ and is given by

$$\mathbf{C}_{\alpha,\beta}(u,v) = u^\alpha v^\beta \cdot \mathbf{C}(u^{1-\alpha},v^{1-\beta})\mbox{.}$$

The composite1COP function provides the means for inserting permutation asymmetry from a permutation symmetric copula as described by Joe (2017, p. 124), but do so in a more general way through the provision of two and not just one parameter. Joe's description is supported herein if one of the $\alpha$ or $\beta$ is held at zero. Very loosely, the $\alpha > 0$ kicks probability density down towards the lower right corner, whereas $\beta > 0$ kicks density up towards the upper left corner. Finally, the composite2COP function is based on a slighty more general result (see composite2COP for further details of copula composition).

Usage

composite1COP(u, v, para, ...)

Value

Value(s) for the composited copula are returned.

Arguments

u: Nonexceedance probability $u$ in the $X$ direction;
v: Nonexceedance probability $v$ in the $Y$ direction;
para: A special parameter list (see Note); and
...: Additional arguments to pass to the copula.

Author

W.H. Asquith

References

Joe, H., 2017, Parametric copula families for statistical models (chap. 8) in Copulas and dependence models with applications---Contributions in honor of Roger B. Nelsen, eds. Flores, U.M., Amo Artero, E., Durante, F., Sánchez, J.F.: Springer, Cham, Switzerland, ISBN 978--3--319--64220--9, tools:::Rd_expr_doi("10.1007/978-3-319-64221-5").