wolfCOP: The Schweizer and Wolff's Sigma of a Copula

Description

Compute the measure of association known as Schweizer and Wolff's Sigma $\sigma_\mathbf{C}$ of a copula according to Nelsen (2006, p. 209) by $$\sigma_\mathbf{C} = 12\int\!\!\int_{\mathcal{I}^2} |\mathbf{C}(u,v) - uv|\,\mathrm{d}u\mathrm{d}v\mbox{.}$$ It is obvious that this measure of association is similar to the following form of Spearman's Rho $$\rho_\mathbf{C} = 12\int\!\!\int_{\mathcal{I}^2} [\mathbf{C}(u,v) - uv]\,\mathrm{d}u\mathrm{d}v\mbox{,\ and}$$ if a copula is positively quadrant dependent (PQD, see isCOP.PQD) then $\sigma_\mathbf{C} = \rho_\mathbf{C}$ and conversely if a copula is negatively quadrant dependent (NQD) then $\sigma_\mathbf{C} = -\rho_\mathbf{C}$. However, a feature that makes $\sigma_\mathbf{C}$ especially attractive is that random variables $X$ and $Y$ that are not PQD or NQD---that is copulas that are neither larger nor smaller than $\mathbf{\Pi}$---is that $\sigma_\mathbf{C}$ is often a better measure of [dependency] than $\rho_\mathbf{C}$ (Nelsen, 2006, p. 209).

Usage

wolfCOP(cop=NULL, para=NULL, brute=FALSE, delta=0.002, ...)

Arguments

cop

A copula function;

para

Vector of parameters or other data structure, if needed, to pass to the copula;

brute

Should brute force be used instead of two nested integrate() functions in Rto perform the double integration;

delta

The $\mathrm{d}u$ and $\mathrm{d}v$ for the brute force (brute=TRUE) integration; and

...

Additional arguments to pass.

Value

The value for $\sigma_\mathbf{C}$ is returned.

References

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

Examples

Run this code

wolfCOP(cop=PSP)

# EXTENDED EXAMPLE --- with ## Not run: to be added by user for more graphics.
para <- list(cop1=PLACKETTcop, cop2=PLACKETTcop, para1=0.145, para2=21.9,
             alpha=0.81,       beta=0.22)
D <- simCOP(n=250, cop=composite2COP, para=para,
            cex=0.5, col=rgb(0,0,0,0.2), pch=16)
# Globally PQD but there is a significant local NQD part of IxI space that is
PQD <- isCOP.PQD(cop=composite2COP, para=para, uv=D)
points(D, col=PQD$local.PQD+2, lwd=2)
# clearly NQD, so this copula interacts through the P copula. Hence by the logic
# of Nelsen (2006), then Schweizer-Wolff Sigma should be larger than Spearman Rho.
wolfCOP(cop=composite2COP, para=para) # 0.08373378
 rhoCOP(cop=composite2COP, para=para) # 0.02845131
hoefCOP(cop=composite2COP, para=para) # 0.08563823
# which is clearly present in these simulated data. In fact the output above also shows
# Sigma to be larger than Kendall Tau, Gini Gamma, and Blomqvist Beta thus Sigma has
# captured the fact that although the symbols plot near randomly on the figure, the
# the symbol coloring for PQD and NQD clearly shows local dependency differences.
the.grid <- EMPIRgrid(para=D)
the.persp <- persp(the.grid$empcop, theta=-25, phi=20, shade=TRUE,
                   xlab="U VARIABLE", ylab="V VARIABLE", zlab="COPULA C(u,v)")
empcop <- EMPIRcopdf(para=D) # data.frame of all points
points(trans3d(empcop$u, empcop$v, empcop$empcop, the.persp),  cex=0.7,
       col=rgb(0,1-sqrt(empcop$empcop),1,sqrt(empcop$empcop)), pch=16)
points(trans3d(empcop$u, empcop$v, empcop$empcop, the.persp),
       col=PQD$local.PQD+1, pch=1)

layout(matrix(c(1,2,3,4), 2, 2, byrow = TRUE), respect = TRUE)
PQD.NQD.cop <- gridCOP(cop=composite2COP, para=para)
Pi <- gridCOP(cop=P)
RHO <- PQD.NQD.cop - Pi; SIG <- abs(RHO)
the.persp <- persp(PQD.NQD.cop, theta=-25, phi=20, shade=TRUE, cex=0.5,
               xlab="U VARIABLE", ylab="V VARIABLE", zlab="COPULA C(u,v)")
mtext("The Copula that has local PQD and NQD", cex=0.5)
the.persp <- persp(Pi, theta=-25, phi=20, shade=TRUE, cex=0.5,
               xlab="U VARIABLE", ylab="V VARIABLE", zlab="COPULA C(u,v)")
mtext("Independence (Pi)", cex=0.5)
the.persp <- persp(RHO, theta=-25, phi=20, shade=TRUE, cex=0.5,
               xlab="U VARIABLE", ylab="V VARIABLE", zlab="COPULA C(u,v)")
mtext("Copula delta: Integrand of Spearman's Rho", cex=0.5)
the.persp <- persp(SIG, theta=-25, phi=20, shade=TRUE, cex=0.5,
               xlab="U VARIABLE", ylab="V VARIABLE", zlab="COPULA C(u,v)")
mtext("abs(Copula delta): Integrand of Schweizer-Wolff's Sigma", cex=0.5)