RFcop: 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

Nonexceedance probability $u$ in the $X$ direction;

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 $\Theta$, and

rho

Spearman Rho if the rho is given.

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.

References

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

Examples

Run this code

# NOT RUN {
# 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)#
# }