FRECHETcop: The Fréchet Family Copula

Description

The Fréchet Family copula (Durante, 2007, pp. 256--259) is $$\mathbf{C}_{\alpha, \beta}(u,v) = \mathbf{FF}(u,v) = \alpha\mathbf{M}(u,v) + (1-\alpha-\beta)\mathbf{\Pi}(u,v)+\beta\mathbf{W}(u,v)\mbox{,}$$ where $\alpha, \beta \ge 0$ and $\alpha + \beta \le 1$. The Fréchet Family copulas are convex combinations of the fundamental copulas $\mathbf{W}$ (Fréchet--Hoeffding lower-bound copula; W), $\mathbf{\Pi}$ (independence; P), and $\mathbf{M}$ (Fréchet--Hoeffding upper-bound copula; M). The copula is comprehensive because both $\mathbf{W}$ and $\mathbf{M}$ can be obtained. The parameters are readily estimated using Spearman Rho ($\rho_\mathbf{C}$; rhoCOP) and Kendall Tau ($\tau_\mathbf{C}$; tauCOP) by

$$\tau_\mathbf{C} = \frac{(\alpha - \beta)(\alpha + \beta + 2)}{3}\mbox{\ and\ } \rho_\mathbf{C} = \alpha - \beta\mbox{.}$$

The Fréchet Family copula virtually always has a visible singular component unless $\alpha, \beta = 0$. The copula has respective lower- and upper-tail dependency parameters of $\lambda^L = \alpha$ and $\lambda^U = \alpha$ (taildepCOP). Durante (2007, p. 257) reports that the Fréchet Family copula can approximate any bivariate copula in a “unique way” and the error bound can be estimated.

Usage

FRECHETcop(u,v, para=NULL, rho=NULL, tau=NULL, par2rhotau=FALSE, ...)

Value

Value(s) for the copula are returned using the $\alpha$ and $\beta$ as set by argument para; however, if para=NULL and rho and tau are set and compatible with the copula, then $\{\rho_\mathbf{C}, \tau_\mathbf{C}\} \rightarrow \{\alpha, \beta\}$, parameter estimation made, and an R

list is returned.

Arguments

u: Nonexceedance probability $u$ in the $X$ direction;
v: Nonexceedance probability $v$ in the $Y$ direction;
para: A vector (two element) of parameters $\alpha$ and $\beta$;
rho: Spearman Rho from which to estimate the parameters;
tau: Kendall Tau from which to estimate the parameters;
par2rhotau: A logical that if TRUE will return an R list of the $\rho_\mathbf{C}$ and $\tau_\mathbf{C}$ for the parameters; and
...: Additional arguments to pass.

Author

W.H. Asquith

Details

The function will check the consistency of the parameters whether given by argument or computed from $\rho_\mathbf{C}$ and $\tau_\mathbf{C}$. The term “Family” is used with this particular copula in copBasic so as to draw distinction to the Fréchet lower- and upper-bound copulas as the two limiting copulas are called.

For no other reason than that it can be easily done and makes a nice picture, loop through a nest of $\rho$ and $\tau$ for the Fréchet Family copula and plot the domain of the resulting parameters:


  ops <- options(warn=-1) # warning supression because "loops" are dumb
  taus <- rhos <- seq(-1,1, by=0.01)
  plot(NA, NA, type="n", xlim=c(0,1), ylim=c(0,1),
       xlab="Frechet Copula Parameter Alpha",
       ylab="Frechet Copula Parameter Beta")
  for(tau in taus) {
    for(rho in rhos) {
      fcop <- FRECHETcop(rho=rho, tau=tau)
      if(! is.na(fcop$para[1])) points(fcop$para[1], fcop$para[2])
    }
  }
  options(ops)

References

Durante, F., 2007, Families of copulas, Appendix C, in Salvadori, G., De Michele, C., Kottegoda, N.T., and Rosso, R., 2007, Extremes in Nature---An approach using copulas: Springer, 289 p.