Numerically set a logical whether a copula is radially symmetric (Nelsen, 2006, p. 37) [reflection symmetric, Joe (2014, p. 64)]. A copula \(\mathbf{C}(u,v)\) is radially symmetric if and only if for any \(\{u,v\} \in [0,1]\) either of the following hold $$\mathbf{C}(u,v) = u + v - 1 + \mathbf{C}(1-u, 1-v)$$ or $$u + v - 1 + \mathbf{C}(1-u, 1-v) - \mathbf{C}(u,v) \equiv 0\mbox{.}$$
Thus, if the equality of the copula \(\mathbf{C}(u,v) = \hat{\mathbf{C}}(u,v)\) (the survival copula), then radial symmetry exists: COP \(=\) surCOP or \(\mathbf{C}(u,v) = \hat{\mathbf{C}}(1-u,1-v)\). The computation is (can be) CPU intensive.
isCOP.radsym(cop=NULL, para=NULL, delta=0.005, tol=1e-4, ...)A logical TRUE or FALSE is returned.
A copula function;
Vector of parameters, if needed, to pass to the copula;
The increments of \(\{u,v\} \mapsto [0+\Delta\delta, 1-\Delta\delta, \Delta\delta]\);
A tolerance on the check for symmetry, default 1 part in 10,000, which is the test for the \(\equiv 0\) (zero equivalence, see source code); and
Additional arguments to pass to the copula or derivative of a copula function.
W.H. Asquith
Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.
Salvadori, G., De Michele, C., Kottegoda, N.T., and Rosso, R., 2007, Extremes in Nature---An approach using copulas: Springer, 289 p.
isCOP.permsym