wolfCOP: The Schweizer and Wolff Sigma of a Copula

Description

Compute the measure of association known as Schweizer--Wolff Sigma $\sigma_\mathbf{C}$ of a copula according to Nelsen (2006, p. 209) by

$$\sigma_\mathbf{C} = 12\int\!\!\int_{\mathcal{I}^2} |\mathbf{C}(u,v) - uv|\,\mathrm{d}u\mathrm{d}v\mbox{,}$$

which is $0 \le \sigma_\mathbf{C} \le 1$. It is obvious that this measure of association, without the positive sign restriction, is similar to the following form of Spearman's Rho (rhoCOP) of a copula:

$$\rho_\mathbf{C} = 12\int\!\!\int_{\mathcal{I}^2} [\mathbf{C}(u,v) - uv]\,\mathrm{d}u\mathrm{d}v\mbox{.}$$

If a copula is positively quadrant dependent (PQD, see isCOP.PQD) then $\sigma_\mathbf{C} = \rho_\mathbf{C}$ and conversely if a copula is negatively quadrant dependent (NQD) then $\sigma_\mathbf{C} = -\rho_\mathbf{C}$. However, a feature making $\sigma_\mathbf{C}$ especially attractive is that for random variables $X$ and $Y$, which are not PQD or NQD---copulas that are neither larger nor smaller than $\mathbf{\Pi}$---is that “$\sigma_\mathbf{C}$ is often a better measure of [dependency] than $\rho_\mathbf{C}$” (Nelsen, 2006, p. 209).

Usage

wolfCOP(cop=NULL, para=NULL, as.sample=FALSE, brute=FALSE, delta=0.002, ...)

Value

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

Arguments

cop: A copula function;
para: Vector of parameters or other data structure, if needed, to pass to the copula;
as.sample: A logical controlling whether an optional R data.frame in para is used to compute the $\hat{\sigma}_\mathbf{C}$ (see Note). If set to -1, then the message concerning CPU effort will be surpressed;
brute: Should brute force be used instead of two nested integrate() functions in R to perform the double integration;
delta: The $\mathrm{d}u$ and $\mathrm{d}v$ for the brute force (brute=TRUE) integration; and
...: Additional arguments to pass.

Author

W.H. Asquith

References

Póczos, Barnabás, Krishner, Sergey, Pál, Szepesvári, Csaba, and Schneider, Jeff, 2015, Robust nonparametric copula based dependence estimators: Accessed on August 11, 2015 at https://www.cs.cmu.edu/~bapoczos/articles/poczos11nipscopula.pdf.

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