footCOP: The Spearman Footrule of a Copula

Description

Compute the measure of association known as the Spearman Footrule $\psi_\mathbf{C}$ (Nelsen et al., 2001, p. 281), which is defined as

$$\psi_\mathbf{C} = \frac{3}{2}\mathcal{Q}(\mathbf{C},\mathbf{M}) - \frac{1}{2}\mbox{,}$$

where $\mathbf{C}(u,v)$ is the copula, $\mathbf{M}(u,v)$ is the Fréchet--Hoeffding upper bound (M), and $\mathcal{Q}(a,b)$ is a concordance function (concordCOP) (Nelsen, 2006, p. 158). The $\psi_\mathbf{C}$ in terms of a single integration pass on the copula is $$\psi_\mathbf{C} = 1 - \int_{\mathcal{I}^2} |u-v|\,\mathrm{d}\mathbf{C}(u,v) = 6 \int_0^1 \mathbf{C}(u,u)\,\mathrm{d}u - 2\mbox{.}$$ Note, Nelsen et al. (2001) use $\phi_\mathbf{C}$ but that symbol is taken in copBasic for the Hoeffding Phi (hoefCOP), and Spearman Footrule does not seem to appear in Nelsen (2006). From the definition, Spearman Footrule only depends on the primary diagnonal (alt. main diagonal, Genest et al., 2010) of the copula, $\mathbf{C}(t,t)$ (diagCOP).

Usage

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

Value

The value for $\psi_\mathbf{C}$ 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 integral to compute $\psi_\mathbf{C}$, use the concordance function method implemented through concordCOP; and
as.sample: A logical controlling whether an optional R data.frame in para is used to compute the $\hat\psi$ (see Note); and
...: Additional arguments to pass, which are dispatched to the copula function cop and possibly concordCOP, such as brute or 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, tools:::Rd_expr_doi("10.1080/10485250903499667").

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