FRECHETcop {copBasic}

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, ...)

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.

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)

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.

Note

A convex combination (convex2COP) of \mathbf{\Pi} and \mathbf{M}, which is a modification of the Fréchet Family, is the Linear Spearman copula:

\mathbf{C}_\alpha(u,v) = (1-\alpha)\mathbf{\Pi}(u,v) + \alpha\mathbf{M}(u,v)\mbox{,}

for 0 \le \alpha \le 1, and the parameter is equal to \rho_\mathbf{C}. When the convex combination is used for construction, the complement of the parameter is equal to \rho_\mathbf{C} (e.g. 1-\alpha = \rho_\mathbf{C}; rhoCOP), which can be validated by

  rhoCOP(cop=convex2COP, para=list(alpha=1-0.48, cop1=P, cop2=M)) # 0.4799948

Author(s)

W.H. Asquith

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.

See Also

M, P, W

Examples

## Not run: 
ppara <- c(0.25, 0.50)
fcop <- FRECHETcop(para=ppara, par2rhotau=TRUE)
RHO <- fcop$rho; TAU <- fcop$tau

level.curvesCOP(cop=FRECHETcop, para=ppara) # Durante (2007, Fig. C.27(b))
mtext("Frechet Family copula")
 UV <- simCOP(n=50, cop=FRECHETcop, para=ppara, ploton=FALSE, points=FALSE)
tau <- cor(UV$U, UV$V, method="kendall" ) # sample Kendall Tau
rho <- cor(UV$U, UV$V, method="spearman") # sample Spearman Rho
spara <- FRECHETcop(rho=rho, tau=tau) # a fitted Frechet Family copula
spara <- spara$para
if(is.na(spara[1])) { # now a fittable combination is not guaranteed
   warning("sample rho and tau do not provide valid parameters, ",
           "try another simulation")
} else { # now if fit, draw some red-colored level curves for comparison
   level.curvesCOP(cop=FRECHETcop, para=spara, ploton=FALSE, col=2)
} #
## End(Not run)