giniCOP: The Gini's Gamma of a Copula

Description

Compute the Gini's Gamma $\gamma_\mathbf{C}$ of a copula according to Nelson (2006) by

$$\gamma_\mathbf{C} = Q(\mathbf{C},\mathbf{M}) + Q(\mathbf{C},\mathbf{W})\mbox{,}$$

where $\mathbf{C}(u,v)$ is the copula, $\mathbf{M}(u,v)$ is M, and $\mathbf{W}(u,v)$ is W. The function $Q(a,b)$ is a concordance function (Nelson, 2006, p.~158), which is implemented by a special call to tauCOP.

Nelson reports that Gini's Gamma measures a concordance relation of 'distance' between $\mathbf{C}(u,v)$ and monotone dependence, as represented by the copulas $\mathbf{M}(u,v)$ and $\mathbf{W}(u,v)$.

The simpler method of computation and the default for the function here, is to compute Gini's $\gamma_\mathbf{C}$ by

$$\gamma_\mathbf{C} = 4\biggl[\int_0^1 \mathbf{C}(u,u) \mathrm{d}u + \int_0^1 \mathbf{C}(u,1-u) \mathrm{d}u\biggr] - 2\mbox{.}$$ This second method is simpler because the single integration is readily deployed (and fast) using two separate calls to the integrate() function of R.

Usage

giniCOP(cop=NULL, para=NULL, byQ=FALSE, delta=0.002, ...)

Arguments

cop

A copula function;

para

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

byQ

Instead of using the single integrals (p.~181~and~182) and the integrate() function to compute $\gamma_\mathbf{C}$, use the concordance function method implemented through tauCOP;

delta

The $\mathrm{d}u$ and $\mathrm{d}v$ for the double integration performed by tauCOP; and

...

Additional arguments to pass, which are dispatched to the copula function cop and possibly tauCOP if byQ=TRUE.

Value

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

References

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

Examples

Run this code

gamma1 <- giniCOP(cop=PSP)
print(gamma1)

gamma2 <- giniCOP(cop=PSP, byQ=TRUE)
print(gamma2)