rhoCOP: The Spearman Rho of a Copula

Description

Compute the measure of association known as the Spearman Rho $\rho_\mathbf{C}$ of a copula according to Nelsen (2006, pp. 167--170, 189, 208) by $$\rho_\mathbf{C} = 12\int\!\!\int_{\mathcal{I}^2} \mathbf{C}(u,v)\, \mathrm{d}u\mathrm{d}v - 3\mbox{,}$$ or $$\rho_\mathbf{C} = 12\int\!\!\int_{\mathcal{I}^2} [\mathbf{C}(u,v) - uv]\, \mathrm{d}u\mathrm{d}v\mbox{,}$$ where the later equation is implemented by rhoCOP as the default method (method="default"). This equation, here having $p = 1$ and $k_p(1) = 12$, is generalized under hoefCOP. The absence of the $12$ in the above equation makes it equal to the covariance defined by the Hoeffding Identity (Joe, 2014, p. 54): $$\mathrm{cov}(U, V) = \int\!\!\int_{\mathcal{I}^2} [\mathbf{C}(u,v) - uv]\, \mathrm{d}u\mathrm{d}v\mbox{ or}$$ $$\mathrm{cov}(U, V) = \int\!\!\int_{\mathcal{I}^2} [\hat{\mathbf{C}}(u,v) - uv]\, \mathrm{d}u\mathrm{d}v\mbox{, which is}$$ $$\mathrm{cov}(U, V) = \int\!\!\int_{\mathcal{I}^2} [u+v-1+\mathbf{C}(1-u,1-v) - uv]\, \mathrm{d}u\mathrm{d}v\mbox{.}$$

Depending on copula family (Joe, 2014, pp. 56 and 267), the alternative formulation for $\rho_\mathbf{C}$ could be used $$\rho_\mathbf{C} = 3 - 12\int\!\!\int_{\mathcal{I}^2} u \frac{\delta\mathbf{C}(u,v)}{\delta u} \, \mathrm{d}u\mathrm{d}v = 3 - 12\int\!\!\int_{\mathcal{I}^2} v\frac{\delta\mathbf{C}(u,v)}{\delta v} \, \mathrm{d}u\mathrm{d}v\mbox{,}$$ where the first integral form corresponds to Joe (2014, eq. 248, p. 56) and is the method="joe21", and the second integral form is the method="joe12".

The integral $$\int\!\!\int_{\mathcal{I}^2} \mathbf{C}(u,v)\,\mathrm{d}u\mathrm{d}v\mbox{,}$$ represents the “volume under the graph of the copula and over the unit square” (Nelsen, 2006, p. 170) and therefore $\rho_\mathbf{C}$ is simple a rescaled volume under the copula. The second equation for $\rho_\mathbf{C}$ expresses the “average distance” between the joint distribution and statistical independence $\mathbf{\Pi} = uv$. Nelsen (2006, pp. 175--176) shows that the following relation between $\rho_\mathbf{C}$ and $\tau_\mathbf{C}$ (tauCOP) exists $$-1 \le 3\tau - 2\rho \le 1\mbox{.}$$

Usage

rhoCOP(cop=NULL, para=NULL, method=c("default", "joe21", "joe12"),
                            as.sample=FALSE, brute=FALSE, delta=0.002, ...)

Value

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

Arguments

cop: A copula function;
para: Vector of parameters or other data structure, if needed, to pass to the copula;
method: The form of integration used to compute (see above);
as.sample: A logical controlling whether an optional R data.frame in para is used to compute the $\hat\rho$ by dispatch to cor() function in R with method = "spearman";
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 integration using brute; and
...: Additional arguments to pass.

Author

W.H. Asquith

References

Joe, H., 2014, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p.