Compute the measure of association known as the Hoeffding Phi \(\Phi_\mathbf{C}\) of a copula from independence (\(uv = \mathbf{\Pi}\); P) according to Cherunbini et al. (2004, p. 164) by
$$\Phi_\mathbf{C} = 3 \sqrt{10\int\!\!\int_{\mathcal{I}^2} \bigl(\mathbf{C}(u,v) - uv\bigr)^2\,\mathrm{d}u\mathrm{d}v}\mbox{,}$$
and Nelsen (2006, p. 210) shows this as
$$\Phi_\mathbf{C} = \biggl(90\int\!\!\int_{\mathcal{I}^2} |\mathbf{C}(u,v) - uv|^2\,\mathrm{d}u\mathrm{d}v\biggr)^{1/2}\mbox{,}$$
for which \(\Phi^2_\mathbf{C}\) (the square of the quantity) is known as the dependence index. Gai<U+00DF>er et al. (2010, eq. 1) have \(\Phi^2_\mathbf{C}\) as the Hoeffding Phi-Square, and their definition, when square-rooted, matches Nelsen's listing.
A generalization (Nelsen, 2006) to \(L_p\) distances from independence (\(uv = \mathbf{\Pi}\); P) through the LpCOP function is
$$L_p \equiv \Phi_\mathbf{C}(p) = \biggl(k(p)\int\!\!\int_{\mathcal{I}^2} |\mathbf{C}(u,v) - uv|^p\,\mathrm{d}u\mathrm{d}v\biggr)^{1/p}\mbox{,}$$
for a \(p: 1 \le p \le \infty\) and where \(k(p)\) is a normalization constant such that \(\Phi_\mathbf{C}(p) = 1\) when the copula \(\mathbf{C}\) is \(\mathbf{M}\) (see M) or \(\mathbf{W}\) (see W). The \(k(p)\) (bivariate definition only) for other powers is given (Nelsen, 2006, exer. 5.44, p. 213) in terms of the complete gamma function \(\Gamma(t)\) by
$$k(p) = \frac{\Gamma(2p+3)}{2[\Gamma(p + 1)]^2}\mbox{,}$$
which is implemented by the hoefCOP function. It is important to realize that the \(L_p\) distances are all symmetric nonparametric measures of dependence (Nelsen, 2006, p. 210). These are symmetric because distance from independence is used as evident by “\(uv\)” in the above definitions.
Reflection/Radial and Permutation Asymmetry---Asymmetric forms similar to the above distances exist. Joe (2014, p. 65) shows two measures of bivariate reflection asymmetry or radial asymmetry (term favored in copBasic) as the distance between \(\mathbf{C}(u,v)\) and the survival copula \(\hat\mathbf{C}(u,v)\) (surCOP) measured by
$$L_\infty^{(\mathrm{radsym})} = \mathrm{sup}_{0\le u,v\le1}|\mathbf{C}(u,v) - \hat\mathbf{C}(u,v)|\mbox{,}$$
or its \(L_p^{(\mathrm{radsym})}\) counterpart
$$L_p^{(\mathrm{radsym})} = \biggl[\int\!\!\int_{\mathcal{I}^2} |\mathbf{C}(u,v) - \hat\mathbf{C}(u,v)|^p\,\mathrm{d}u\mathrm{d}v\biggr]^{1/p}\mbox{with\ } p \ge 1\mbox{,}$$
where \(\hat\mathbf{C}(u,v) = u + v - 1 + \mathbf{C}(1-u, 1-v)\) and again \(p: 1 \le p \le \infty\). Joe (2014) does not seem to discuss and normalization constants for these two radial asymmetry distances.
Joe (2014, p. 66) offers analogous measures of bivariate permutation asymmetry (\(\mathbf{C}(u,v) \not= \mathbf{C}(v,u)\)) defined as
$$L_\infty^{(\mathrm{permsym})} = \mathrm{sup}_{0\le u,v\le1}|\mathbf{C}(u,v) - \hat\mathbf{C}(v,u)|\mbox{,}$$
or its \(L_p^{(\mathrm{permsym})}\) counterpart
$$L_p^{(\mathrm{permsym})} = \biggl[\int\!\!\int_{\mathcal{I}^2} |\mathbf{C}(u,v) - \hat\mathbf{C}(v,u)|^p\,\mathrm{d}u\mathrm{d}v\biggr]^{1/p}\mbox{with\ } p \ge 1\mbox{,}$$
where \(p: 1 \le p \le \infty\). Again, Joe (2014) does not seem to discuss and normalization constants for these two permutation symmetry distances. Joe (2014, p. 65) states that the “simplest one-parameter bivariate copula families [and] most of the commonly used two-parameter bivariate copula families are permutation symmetric.”
The asymmetrical \(L_\infty\) and \(L_p\) measures identified by Joe (2014, p. 66) are nonnegative with an upper bounds that depends on \(p\). The bound dependence on \(p\) is caused by the lack of normalization constant \(k(p)\). In an earlier paragraph, Joe (2014) indicates an upper bounds of 1/3 for both (likely?) concerning \(L_\infty^{(\mathrm{radsym})}\) and \(L_\infty^{(\mathrm{permsym})}\). The numerical integrations for \(L_p^{(\mathrm{radsym})}\) and \(L_p^{(\mathrm{permsym})}\) can readily return zeros and often inspection of the formula for the \(\mathbf{C}(u,v)\) itself would be sufficient to judge whether symmetry exists and hence the distances are uniquely zero.
Joe (2014, p. 66) completes the asymmetry discussion with three definitions of skewness of combinations of random variables \(U\) and \(V\): Two definitions are in uvlmoms (for \(U + V - 1\) and \(U - V\)) and two are for \(V-U\) (nuskewCOP) and \(U+V-1\) (nustarCOP).