prod2COP: The Product of Two Copulas

Description

Perform copula multiplication (so-called “$\ast$-product” or Markov Product) (Darsow and others, 1992) is a continuous analog of matrix multiplication and yields another copula: $$\bigl(\mathbf{C}_1 \ast \mathbf{C}_2 \bigr)(u,v) = \mathbf{C}_3(u,v) = \int_\mathcal{I} \frac{\delta \mathbf{C}_1(u, t)}{\delta v} \frac{\delta \mathbf{C}_2(t, v)}{\delta u}\,\mathrm{d}t\mbox{,}$$ for copulas $\mathbf{C}_1(u, v)$ and $\mathbf{C}_2(u, v)$ are copulas whose $\ast$-product yields copula $\mathbf{C}_3(u, v)$ in terms of partial derivatives (derCOP and derCOP2) of the other two. Nelsen (2006, p. 245) lists several identities of the $\ast$-product involving the product ($\mathbf{\Pi}$; P), lower bound ($\mathbf{W}$; W), and upper bound ($\mathbf{M}$; M) copulas: $$\mathbf{\Pi} \ast \mathbf{C} = \mathbf{C} \ast \mathbf{\Pi} = \mathbf{\Pi}\mbox{,}$$ $$\mathbf{M} \ast \mathbf{C} = \mathbf{C} \ast \mathbf{M} = \mathbf{M}\mbox{,}$$ $$\bigl(\mathbf{W} \ast \mathbf{C}\bigr)(u,v) = v - \mathbf{C}(1-u, v)\mbox{\ and\ } \bigl(\mathbf{C} \ast \mathbf{W}\bigr)(u,v) = u - \mathbf{C}(u, 1-v)\mbox{, and}$$ $$\mathbf{W} \ast \mathbf{W} = \mathbf{M}\mbox{ and } \mathbf{W} \ast \mathbf{C} \ast \mathbf{W} = \hat{\mathbf{C}}\mbox{,}$$ where $\hat{\mathbf{C}}$ is the survival copula (surCOP). The $\ast$-product is associative: $$\mathbf{A} \ast (\mathbf{B} \ast \mathbf{C}) = (\mathbf{A} \ast \mathbf{B}) \ast \mathbf{C}\mbox{,}$$ but $\ast$-product is not commutative (order independent). Nelsen (2006, p. 245) reports that “if we view $\ast$ as a binary operation on the set of copulas, then $\mathbf{\Pi}$ is the null element, and $\mathbf{M}$ is the identity.” Copula mulitiplication is closely linked to Markov Processes (Nelsen, 2006, pp. 244--248).

For other descriptions and computations of copula combination are possible using the copBasic package, see convexCOP, convex2COP, composite1COP, composite2COP, composite3COP, glueCOP, and convexCOP.

Usage

prod2COP(u,v, cop1=NULL, para1=NULL, cop2=NULL, para2=NULL, para=NULL,
              pinterval=NULL, ...)

Value

Value(s) for the copula are returned.

Arguments

u: Nonexceedance probability $u$ in the $X$ direction;
v: Nonexceedance probability $v$ in the $Y$ direction;
cop1: The $\mathbf{C}_1(u,v; \Theta_1)$ copula function with vectorization as in asCOP;
para1: Vector of parameters or other data structures for $\Theta_1$, if needed, to pass to copula $\mathbf{C}_1(u,v; \Theta_1)$;
cop2: The $\mathbf{C}_2(u,v; \Theta_2)$ copula function with vectorization as in asCOP;
para2: Vector of parameters or other data structures for $\Theta_2$, if needed, to pass to copula $\mathbf{C}_2(u,v; \Theta_2)$;
para: An R list that can take the place of the cop1, para1, cop2, and para2 arguments. These four will be populated from same named elements of the list, and if the other four arguments were specified through the function interface, these are silently ignored;
pinterval: An optional interval for the above integral. The default is $\mathcal{I} = [0,1]$ but the option of the user to replace exact end points with “small” numbers is possible (e.g. interval=c(lo, 1-lo) for say lo=.Machine$double.eps). This interval is uniquely picked up for the interval in the above definition of prod2COP. The pinterval can also be set within the para and the function will pick it up from there; and
...: Additional arguments to pass to the copulas.

Author

W.H. Asquith

References

Darsow, W.F., Nguyen, B., and Olsen, E.T., 1992, Copulas and Markov processes: Illinois Journal of Mathematics, v. 26, pp. 600--624, tools:::Rd_expr_doi("10.1215/IJM/1255987328").