Compute the coordinates of the bivariate marginal probabilities for variables \(U\) and \(V\) given selected probabilities levels \(t\) for a copula \(\mathbf{C}(u,v)\) for \(u\) with respect to \(v\). For the case of a joint and probability, symbolically the solution is
$$\mathrm{Pr}[U \le v,\ V \le v] = t = \mathbf{C}(u,v)\mbox{,}$$
where \(V \mapsto [t_i, t_{j}, t_{j+1}, \cdots, 1; \Delta]\) (an irregular sequence of \(v\) values from the \(i\)th value of \(t_i\) provided through to unity) and thus
$$t_i \mapsto \mathbf{C}(u, v=V)\mbox{,}$$
and solving for the sequence of \(u\). The index \(j\) is to indicate that a separate loop is involved and is distinct from \(i\). The pairings \(\{u(t_i), v(t_i)\}\) for each \(t\) are packaged as an R data.frame
. This operation is very similiar to the plotting capabilities in level.curvesCOP2
for level curves (Nelsen, 2006, pp. 12--13) but implemented in the function joint.curvesCOP2
for alternative utility.
For the case of a joint or probability, the dual of a copula (function) or \(\tilde{\mathbf{C}}(u,v)\) from a copula (Nelsen, 2006, pp. 33--34) is used and symbolically the solution is:
$$\mathrm{Pr}[U \le v \mathrm{\ or\ } V \le v] = t = \tilde{\mathbf{C}}(u,v) = u + v - \mathbf{C}(u,v)\mbox{,}$$
where \(V \mapsto [0, v_j, v_{j+1}, \cdots, t_i; \Delta]\) (an irregular sequence of \(v\) values from zero through to the \(i\)th value of \(t\)) and thus
$$t_i \mapsto \tilde{\mathbf{C}}(u, v=V)\mbox{,}$$
and solving for the sequence of \(u\). The index \(j\) is to indicate that a separate loop is involved and is distinct from \(i\). The pairings \(\{u(t_i), v(t_i)\}\) for each \(t\) are packaged as an R data.frame
.