copBasic (version 1.7.1)

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 $\Theta$, $\mathbf{C} \ne \mathbf{\Pi}$ (for $\mathbf{\Pi}$ see P) then a family of generally asymmetric copulas $\mathbf{C}_{\alpha,\beta; \Theta}$ with two compositing parameters $0 < \alpha,\beta < 1, \alpha \ne \beta$ that 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 composite2COP function is based on a slighty more general result (see composite2COP for more details of copula composition).

Usage

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

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.

Value

• Value(s) for the composited copula are returned.

References

Salvadori, G., De Michele, C., Kottegoda, N.T., and Rosso, R., 2007, Extremes in Nature---An approach using copulas: Springer, 289 p.

See Also

COP, composite2COP, composite3COP

Examples

alpha <- 0.24; beta <- 0.23; Theta1 <- NA;
# W() does not use a parameter, but show how a parameter would be set if be needed.
para   <- list(alpha=alpha, beta=beta, cop1=W, para1=Theta1)
t <- composite1COP(0.4,0.6,para)
if( t != W(0.4,0.6)) message("Not equal as expected")