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 the 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 $\rho$ and $\tau$ 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.