composite3COP: (Extended) Composition of Two Copulas with Four Compositing...

In wasquith/copBasic: General Bivariate Copula Theory and Many Utility Functions

(Extended) Composition of Two Copulas with Four Compositing Parameters

Description

The (extended) composition of two copulas (Salvadori et al., 2006, p. 266, prop. C.4) provides for even more sophisticated structures of dependence between variables than two-copula composition in composite2COP. Let \mathbf{A} and \mathbf{B} be copulas with respective parameters \Theta_\mathbf{A} and \Theta_\mathbf{B}, then

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

defines a family of copulas \mathbf{C}_{\alpha,\beta,\kappa,\gamma} with four compositing parameters \alpha,\beta,\kappa,\gamma \in (0,1).

It is important to stress that copulas \mathbf{A}_{\Theta_A} and \mathbf{B}_{\Theta_B} can be of different families and each parameterized accordingly by the vectors of parameters \Theta_A and \Theta_B.

Usage

composite3COP(u, v, para, ...)

Arguments

u	Nonexceedance probability u in X direction;
v	Nonexceedance probability v in Y direction;
para	A special parameter list (see Note); and
...	Additional arguments to pass to composite2COP.

Value

A value for the composited copula is returned.

Note

The following descriptions list in detail the structure and content of the para argument:

alpha

— The \alpha compositing parameter;

beta

— The \beta compositing parameter;

kappa

— The \kappa compositing parameter;

gamma

— The \gamma compositing parameter;

cop1

— Function of the first copula \mathbf{A};

cop2

— Function of the second copula \mathbf{B};

para1

— Vector of parameters \Theta_\mathbf{A} for \mathbf{A}; and

para2

— Vector of parameters \Theta_\mathbf{B} for \mathbf{B}.

The first example produces two plots. These are extremely informative for many nuances of copula theory. Whereas it is difficult in prose to describe, users are strongly encouraged that once full understanding of connection of red and green between the easier to understand bivariate plot and the plot showing the sections and derivatives of the sections is achieved that much of copula theory will be mastered—get a copy of Nelsen (2006) and (or) Salvadori et al. (2007).

Author(s)

W.H. Asquith

References

Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.

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, breveCOP, simCOP, composite1COP, composite2COP, convexCOP, glueCOP, simcomposite3COP

Examples

## Not run: 
para <- list(cop1=PLACKETTcop, cop2=N4212cop,
             para1=10^(runif(1,min=-5,max=5)), para2=runif(1,min=1,max=100),
             alpha=runif(1), beta=runif(1), kappa=runif(1), gamma=runif(1))
txts <- c("Alpha=",    round(para$alpha,    digits=4),
          "; Beta=",   round(para$beta,     digits=4),
          "; Kappa=",  round(para$kappa,    digits=4),
          "; Gamma=",  round(para$gamma,    digits=4),
          "; Theta1=", round(para$para1[1], digits=5),
          "; Theta2=", round(para$para2[1], digits=2))
layout(matrix(1:2, byrow=TRUE))
D <- simCOP(n=300, cop=composite3COP, para=para, cex=0.5, col=rgb(0,0,0,0.2), pch=16)
mtext(paste(txts,collapse=""))

f <- round(runif(1),digits=2)
ftxt <- c("Sectionals (thick) and derivatives (thin) at f=",f," nonexceedance prob.")
segments(f,0,f,1, col=3, lwd=2); segments(0,f,1,f, col=2, lwd=2)
t <- sectionCOP(f,cop=composite3COP,para=para, col=3, lwd=4)
t <- sectionCOP(f,cop=composite3COP,para=para, dercop=TRUE, ploton=FALSE,col=3)
t <- sectionCOP(f,cop=composite3COP,para=para, wrtV=TRUE,   ploton=FALSE,col=2,lwd=4)
t <- sectionCOP(f,cop=composite3COP,para=para, wrtV=TRUE,   ploton=FALSE,col=2,
                  dercop=TRUE)
mtext(paste(ftxt, collapse=""))#
## End(Not run)

wasquith/copBasic documentation built on Feb. 17, 2025, 11:39 a.m.