RFcop {copBasic}

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}\bigl(uv\bigr)^{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). 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).

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, 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.

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)

## Not run: 
set.seed(1)
UV <- simCOP(1000, cop=COP, para=list(cop=RFcop, para=RFcop(rho=0.5)$para, reflect=3))
cor(UV$U, UV$V, method="spearman") # -0.492677 as expected with reversal of V #
## End(Not run)