giniCOP: The Gini Gamma of a Copula

Description

Compute the measure of association known as the Gini Gamma $\gamma_\mathbf{C}$ (Nelsen, 2006, pp. 180--182), which is defined as $$\gamma_\mathbf{C} = \mathcal{Q}(\mathbf{C},\mathbf{M}) + \mathcal{Q}(\mathbf{C},\mathbf{W})\mbox{,}$$ where $\mathbf{C}(u,v)$ is the copula, $\mathbf{M}(u,v)$ is the M function, and $\mathbf{W}(u,v)$ is the W function. The function $\mathcal{Q}(a,b)$ (concordCOP) is a concordance function (Nelsen, 2006, p. 158). Nelsen also reports that “Gini Gamma measures a concordance relation of “distance” between $\mathbf{C}(u,v)$ and monotone dependence, as represented by the Fréchet--Hoeffding lower bound and Fréchet--Hoeffding upper bound copulas [$\mathbf{M}(u,v)$, M and $\mathbf{W}(u,v)$, W respectively]”

A simpler method of computation and the default for giniCOP is to compute $\gamma_\mathbf{C}$ by

$$\gamma_\mathbf{C} = 4\biggl[\int_\mathcal{I} \mathbf{C}(u,u)\,\mathrm{d}u + \int_\mathcal{I} \mathbf{C}(u,1-u)\,\mathrm{d}u\biggr] - 2\mbox{,}$$ or in terms of the primary diagonal $\delta(t)$ and secondary diagonal $\delta^\star(t)$ (see diagCOP) by $$\gamma_\mathbf{C} = 4\biggl[\int_\mathcal{I} \mathbf{\delta}(t)\,\mathrm{d}t + \int_\mathcal{I} \mathbf{\delta^\star }(t)\,\mathrm{d}t\biggr] - 2\mbox{.}$$

The simpler method is more readily implemented because single integration is fast. Lastly, Nelsen et al. (2001, p. 281) show that $\gamma_\mathbf{C}$ also is computable by $$\gamma_\mathbf{C} = 2\,\mathcal{Q}(\mathbf{C},\mathbf{A})\mbox{,}$$ where $\mathbf{A}$ is a convex combination (convex2COP, using $\alpha = 1/2$) of the copulas $\mathbf{M}$ and $\mathbf{W}$ or $\mathbf{A} = (\mathbf{M}+\mathbf{W})/2$. However, integral convergence errors seem to trigger occasionally, and the first definition by summation $\mathcal{Q}(\mathbf{C},\mathbf{M}) + \mathcal{Q}(\mathbf{C},\mathbf{W})$ thus is used. The convex combination is demonstrated in the Examples section.

Usage

giniCOP(cop=NULL, para=NULL, by.concordance=FALSE, as.sample=FALSE, ...)

Value

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

Arguments

cop: A copula function;
para: Vector of parameters or other data structure, if needed, to pass to the copula;
by.concordance: Instead of using the single integrals (Nelsen, 2006, pp. 181--182) to compute $\gamma_\mathbf{C}$, use the concordance function method implemented through concordCOP;
as.sample: A logical controlling whether an optional R data.frame in para is used to compute the $\hat\gamma_\mathbf{C}$ (see Note); and
...: Additional arguments to pass, which are dispatched to the copula function cop and possibly concordCOP if by.concordance=TRUE, such as delta used by that function.

Author

W.H. Asquith

References

Genest, C., Nešlehová, J., and Ghorbal, N.B., 2010, Spearman's footrule and Gini's gamma---A review with complements: Journal of Nonparametric Statistics, v. 22, no. 8, pp. 937--954.

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