RFcop: The Raftery Copula

In copBasic: General Bivariate Copula Theory and Many Utility Functions

The Raftery Copula

Description

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}(uv)^{1/(1-\Theta)}\bigl[1-(\mathrm{max}\{u,v\})^{-(1+\Theta)/(1-\Theta)}\bigr]\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).

Usage

RFcop(u, v, para=NULL, rho=NULL, tau=NULL, fit=c("rho", "tau"), ...)

Arguments

u	Nonexceedance probability u in the X direction;
v	Nonexceedance probability v in the Y direction;
para	A vector (single element) of parameters—the \Theta parameter of the copula;
rho	Optional Spearman Rho from which the parameter will be estimated and presence of rho trumps tau;
tau	Optional Kendall Tau from which the parameter will be estimated;
fit	If para, rho, and tau are all NULL, the 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 not other utility than to trigger which measure of association is computed internally by the cor function in R; and
...	Additional arguments to pass.

Value

Value(s) for the copula are returned. Otherwise if either rho or tau is given, then the \Theta is computed and a list having

para	The parameter \Theta;
rho	Spearman Rho if the rho is given; and
tau	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.

Author(s)

W.H. Asquith

References

Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.

See Also

M, P

Examples

# Lower tail dependency of Theta = 0.5 --> 2*(0.5)/(1+0.5) = 2/3 (Nelsen, 2006, p. 214)
taildepCOP(cop=RFcop, para=0.5)$lambdaL # 0.66667

## Not run: 
# Simulate for a Spearman Rho of 0.7, then extract estimated Theta that
# internally is based on Kendall Tau of U and V, then convert estimate
# to equivalent Rho.
set.seed(1)
UV <- simCOP(1000, cop=RFcop, RFcop(rho=0.7)$para)
Theta <- RFcop(UV$U, UV$V, fit="tau")$para # 0.607544
Rho   <- Theta*(4-3*Theta)/(2-Theta)^2     # 0.682255 (nearly 0.7)#
## End(Not run)

copBasic documentation built on Oct. 17, 2023, 5:08 p.m.