composite1COP: Composition of a Single Symmetric Copula with Two Compositing Parameters (Khoudraji Device with Pi Independence)

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. This construction technique is named the Khoudraji device within the copula package (see khoudrajiCopula therein) (Hofert et al., 2018, pp. 120--121). 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 and more contextualization of Hofert et al. (2018) remarks on the Khoudraji device).

Usage

composite1COP(u, v, para, ...)
khoudraji1COP(u, v, para, ...)
khoudrajiPCOP(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

Hofert, M., Kojadinovic, I., Mächler, M., and Yan, J., 2018, Elements of copula modeling with R: Dordrecht, Netherlands, Springer.

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").