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 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.