Compute the copula or joint distribution function through a copula as shown by Nelsen (2006, p. 18) is the joint probability
$$\mathrm{Pr}[U \le u, V \le v] = \mathbf{C}(u,v)\mbox{.}$$
The copula is an expression of the joint probability that both \(U \le u\) and \(V \le v\).
A copula is a type of dependence function that permits straightforward characterization of dependence from independence. Joe (2014, p. 8) comments that “copula families are usually given as cdfs [cumulative distribution functions.]” A radially symmetric or permutation symmetric copula is one such that \(\mathbf{C}(u,v) = \mathbf{C}(v,u)\) otherwise the copula is asymmetric.
The copula inversions \(t = \mathbf{C}(u{=}U, v)\) or \(t = \mathbf{C}(u, v{=}V)\) for a given \(t\) and \(U\) or \(V\) are provided by COPinv and COPinv2, respectively. A copula exists in the domain of the unit square (\(\mathcal{I}^2 = [0, 1]\times [0,1]\)) and is a grounded function meaning that
$$\mathbf{C}(u,0) = 0 = \mathbf{C}(0,v) \mbox{\ and\ thus\ } \mathbf{C}(0,0) = 0\mbox{, }$$
and other properties of a copula are that
$$\mathbf{C}(u,1) = u \mbox{\ and\ } \mathbf{C}(1,v) = v\mbox{\ and}$$
$$\mathbf{C}(1,1) = 1\mbox{.}$$
Copulas can be combined with each other (convexCOP, convex2COP, composite1COP,
composite2COP, composite3COP, and glueCOP) to form more complex and sophisticated dependence structures. Also copula multiplication---a special product of two copulas---yields another copula (see prod2COP).
Perhaps the one of the more useful features of this function is that in practical applications it can be used to take a copula formula and reflect or rotated it in fashions to attain association structures that the native definition of the copula can not acquire. The terminal demonstration in the Examples demonstrates this for the Raftery copula (RFcop).
COP(u, v, cop=NULL, para=NULL,
reflect=c("cop", "surv", "acute", "grave",
"1", "2", "3", "4"), ...)Value(s) for the copula are returned.
Nonexceedance probability \(u\) in the \(X\) direction;
Nonexceedance probability \(v\) in the \(Y\) direction;
A copula function with vectorization as in asCOP;
Vector of parameters or other data structures, if needed, to pass to the copula;
The reflection of the copula form (see Note) and the default "cop" or "1" is the usual copula definition (also see simCOPmicro). The numbered values correspond, respectively, to the named values; and
Additional arguments to pass to the copula.
W.H. Asquith
Joe, H., 2014, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p.
Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.
coCOP, duCOP, surCOP, surfuncCOP