composite2COP: Composition of Two Copulas with Two Compositing Parameters (Khoudraji Device)

Description

The composition of two copulas (Salvadori et al., 2007, p. 266, prop. C.3) provides for more sophisticated structures of dependence between variables than many single parameter copula can provide. Further, asymmetrical copulas are readily obtained from symmetrical copulas. Let $\mathbf{A}$ and $\mathbf{B}$ be copulas with respective parameters $\Theta_\mathbf{A}$ and $\Theta_\mathbf{B}$, then

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

defines a family of copulas $\mathbf{C}_{\alpha,\beta; \Theta_\mathbf{A}, \Theta_\mathbf{B}}$ with two compositing parameters $\alpha,\beta \in \mathcal{I}:[0,1]$. In particular if $\alpha = \beta = 1$, then $\mathbf{C}_{1,1} = \mathbf{A}$, and if $\alpha = \beta = 0$, then $\mathbf{C}_{0,0} = \mathbf{B}$. For $\alpha \ne \beta$, the $\mathbf{C}_{\alpha,\beta}$ is in general asymmetric that is $\mathbf{C}(u,v) \ne \mathbf{C}(v,u)$ for some $(u,v) \in \mathcal{I}^2$. This construction technique is named the Khoudraji device within the copula package (see khoudrajiCopula therein) (Hofert et al., 2018, p. 120).

It is important to stress that copulas $\mathbf{A}_{\Theta_A}$ and $\mathbf{B}_{\Theta_B}$ can be of different families and each copula parameterized accordingly by the vector of parameters $\Theta_A$ and $\Theta_B$. This is an interesting feature in the context of building complex structures when pursuing asymmetric measures of dependency such as the L-comoments. Symmetry of the copula $\mathbf{C}$ is required for the situation that follows, however.

It is possible to simplify the construction of an asymmetric copula from a symmetric copula by the following. Let $\mathbf{C}(u,v)$ be a symmetric copula, $\mathbf{C} \ne \mathbf{\Pi}$ (for $\mathbf{\Pi}$ see P). A family of asymmetric copulas $\mathbf{C}_{\alpha,\beta}$ with two composition parameters $0 < \alpha,\beta < 1, \mbox{and\ } \alpha \ne \beta$ that also includes $\mathbf{C}(u,v)$ as a limiting case and is given by

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

Hofert et al. (2018, p. 121) comment that “from a practical perspective, a useful subset of families constructed from [the] Khoudraji device is obtained” by choosing independence copula ($\mathbf{\Pi}$, P) for one of the copula and that choosing [$\alpha$ or $\beta$] relative close to $1$ produces nonexchangable [see isCOP.permsym] versions of $\mathbf{C}(u,v)$ (meaning $\mathbf{C}(u,v) \ne \mathbf{C}(v,u)$). For the copBasic package, the khoudraji1COP() and khoudrajiPCOP() (the P meaning P) aliases are added for programming clarity for developers desiring to have contrasting copula calls to the three compositing copula: composite1COP, composite2COP(), and composite3COP.

Usage

composite2COP(u, v, para, ...)
khoudraji2COP(u, v, para, ...)

Value

Value(s) for the composited copula is 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 copulas.

Author

W.H. Asquith

References

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