copBasic (version 2.0.1)

footCOP: The Spearman's Footrule of a Copula

Description

Compute the measure of association known as Spearman's 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{Frechet}-Hoeffding upper bound (M), and $\mathcal{Q}(a,b)$ is a concordance function (Nelsen, 2006, p. 158), which is implemented by a specially formed call to concordCOP. The $\psi_\mathbf{C}$ in terms of a single integration pass on the copula is $$\psi_\mathbf{C} = 6 \int_0^1 \mathbf{C}(u,u)\,\mathrm{d}u - 2\mbox{.}$$Note that Nelsen et al. use $\phi_\mathbf{C}$ but that symbol is taken in copBasic for Hoeffding's Phi (hoefCOP), and Spearman's Footrule does not seem to appear in Nelsen (2006).

Usage

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

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 Rdata.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.

Value

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

encoding

utf8

References

Genest, C., Nešlehová{Neslehova}, 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.

Nelsen, R.B., Quesada-Molina, J.J., Rodríguez-Lallena{Rodriguez-Lallena}, J.A., Úbeda-Flores{Ubeda-Flores}, M., 2001, Distribution functions of copulas---A class of bivariate probability integral transforms: Statistics and Probability Letters, v. 54, no. 3, pp. 277--282.

See Also

blomCOP, giniCOP, hoefCOP, rhoCOP, tauCOP, wolfCOP

Examples

footCOP(cop=PSP)                      # 0.3177662
# footCOP(cop=PSP, by.concordance=TRUE) # 0.3178025

n <- 2000; UV <- simCOP(n=n, cop=GHcop, para=2.3, graphics=FALSE)
footCOP(para=UV, as.sample=TRUE)                  # 0.5594364 (sample version)
footCOP(cop=GHcop, para=2.3)                      # 0.5513380 (copula integration)
footCOP(cop=GHcop, para=2.3, by.concordance=TRUE) # 0.5513562 (concordance function)
# where the later issued warnings on the integration