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
$$\psi_\mathbf{C} = 6 \int_0^1 \mathbf{C}(u,u)\,\mathrm{d}u - 2\mbox{.}$$
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).
footCOP(cop=NULL, para=NULL, by.concordance=FALSE, as.sample=FALSE, ...)
The value for \(\psi_\mathbf{C}\) is returned.
A copula function;
Vector of parameters or other data structure, if needed, to pass to the copula;
Instead of using the single integral to compute \(\psi_\mathbf{C}\), use the concordance function method implemented through concordCOP
; and
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.
W.H. Asquith
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.
Nelsen, R.B., Quesada-Molina, J.J., Rodríguez-Lallena, J.A., Úbeda-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.
blomCOP
, giniCOP
, hoefCOP
,
rhoCOP
, tauCOP
, wolfCOP