copBasic (version 1.7.1)

FRECHETcop: The Fréchet{Frechet} Family Copula

Description

The Fréchet{Frechet} 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{Frechet} Family copulas are convex combinations of the fundamental copulas $\mathbf{W}$ (Fréchet{Frechet}-Hoeffding lower bound copula), $\mathbf{\Pi}$ (independence), and $\mathbf{M}$ (Fréchet{Frechet}-Hoeffding upper bound copula). The copula is comprehensive because both $\mathbf{W}$ and $\mathbf{M}$ can be obtained. The parameters are readily estimated using Spearman's Rho ($\rho$) and Kendall's Tau ($\tau$) by $$\tau = \frac{(\alpha - \beta)(\alpha + \beta + 2)}{3}\mbox{\ and\ } \rho = \alpha - \beta\mbox{.}$$ The copula virtually always has a singular component unless $\alpha, \beta = 0$. The Fréchet{Frechet} Family copula has respective lower- and upper-tail dependency parameters of $\lambda_L = \alpha$ and $\lambda_U = \alpha$. Lastly, Durante (2007) reports that the Fréchet{Frechet} 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's $\rho$ from which to estimate the parameters;

tau

Kendall's $\tau$ from which to estimate the parameters;

par2rhotau

A logical that if TRUE will return an Rlist of the $\rho$ and $\tau$ for the parameters; and

...

Additional arguments to pass.

Value

• Value(s) for the copula are returned using the $\alpha$ and $\beta$ as set by argument para. If para=NULL and rho and tau are set and compatible with the copula, then ${\rho, \tau} \rightarrow {\alpha, \beta}$ and an Rlist is returned.

encoding

utf8

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

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's Tau
rho <- cor(UV$U, UV$V, method="spearman") # sample Spearman's 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)
}