The Raftery copula (Nelsen, 2006, p. 172) is
$$\mathbf{C}_{\Theta}(u,v) = \mathbf{RF}(u,v) = \mathbf{M}(u,v) + \frac{1-\Theta}{1+\Theta}\biggl(uv\biggr)^{1/(1-\Theta)}\biggl[1-\bigl(\mathrm{max}\{u,v\}\bigr)^{-(1+\Theta)/(1-\Theta)}\biggr]\mbox{,}$$
where \(\Theta \in (0,1)\). The copula, as \(\Theta \rightarrow 0^{+}\) limits, to the independence coupla (\(\mathbf{P}(u,v)\); P), and as \(\Theta \rightarrow 1^{-}\), limits to the comonotonicity copula (\(\mathbf{M}(u,v)\); M). The parameter \(\Theta\) is readily computed from Spearman Rho (rhoCOP) by \(\rho_\mathbf{C} = \Theta(4-3\Theta)/(2-\Theta)^2\) or from Kendall Tau (tauCOP) by \(\tau_\mathbf{C} = 2\Theta/(3-\Theta)\). However, this copula like others within the copBasic package can be reflected (rotated) at will with the COP abstraction layer to acquire negative or inverse dependency (countermonotonicity) (see the Examples).
RFcop(u, v, para=NULL, rho=NULL, tau=NULL, fit=c("rho", "tau"), ...)Value(s) for the copula are returned. Otherwise if either rho or tau is given, then the \(\Theta\) is computed and a list having
The parameter \(\Theta\);
Spearman Rho if the rho is given; and
Kendall Tau if the tau is given but also if both rho and tau are NULL as mentioned next.
and if para=NULL and rho and tau=NULL, then the values within u and v are used to compute Kendall Tau and then compute the parameter, and these are returned in the aforementioned list.
Nonexceedance probability \(u\) in the \(X\) direction;
Nonexceedance probability \(v\) in the \(Y\) direction;
A vector (single element) of parameters---the \(\Theta\) parameter of the copula;
Optional Spearman Rho from which the parameter will be estimated and presence of rho trumps tau;
Optional Kendall Tau from which the parameter will be estimated;
If para, rho, and tau are all NULL, then the u and v represent the sample. The measure of association by the fit declaration will be computed and the parameter estimated subsequently. The fit has no other utility than to trigger which measure of association is computed internally by the cor function in R; and
Additional arguments to pass.
W.H. Asquith
Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.
M, P