copBasic (version 2.0.1)

rhobevCOP: A Dependence Measure for a Bivariate Extreme Value Copula based on the Expectation of the Product of Negated Log-Transformed Random Variables U and V

Description

Compute a dependence measure based on the expectation of the product of transformed random variables $U$ and $V$, which unnamed by Joe (2014, pp. 383--384) but symbolically is $\rho_E$, having a bivariate extreme value copula $\mathbf{C}_{BEV}(u,v)$ by $$\rho_E = \mathrm{E}[(-\log U) \times (-\log V)] - 1 = \int_0^1 [B(w)]^{-2}\,\mathrm{d}w - 1\mbox{,}$$ where $B(w) = A(w, 1-w)$, $B(0) = B(1) = 1$, $B(w) \ge 1/2$, and $0 \le w \le 1$, and where only bivariate extreme value copulas can be written as $$\mathbf{C}_{BEV}(u,v) = \mathrm{exp}{-A(-\log u, -\log v) }\mbox{,}$$ and thus in terms of the coupla $$B(w) = -\log{\mathbf{C}_{BEV}(\mathrm{exp}[-w], \mathrm{exp}[w-1])}\mbox{.}$$

Joe (2014, p. 383) states that $\rho_E$ is the correlation of the survival function of a bivariate min-stable exponential distribution, which can be assembled as a function of $B(w)$. Joe (2014, p. 383) also shows the following expression for Spearman's Rho $$\rho_S = 12 \int_0^1 [1 + B(w)]^{-2}\,\mathrm{d}w - 3$$ in terms of $B(w)$. This expression inconjunction with rhoCOP was used to confirm the prior expression shown here for $B(w)$ in terms of $\mathbf{C}_{BEV}(u,v)$.

For independence ($uv = \mathbf{\Pi}$; P), $\rho_E = 0$ and for the Fréchet{Frechet}-Hoeffding upper bound copula (perfect positive association), $\rho_E = 1$.

Usage

rhobevCOP(cop=NULL, para=NULL, as.sample=FALSE, brute=FALSE, delta=0.002, ...)

Arguments

cop

A bivariate extreme value copula function---the function rhobevCOP makes no provision for verifying whether the copula in cop is actually an extreme value copula;

para

Vector of parameters or other data structure, if needed, to pass to the copula;

as.sample

A logical controlling whether an optional Rdata.frame in para is used to compute a $\hat\rho_E$ by mean() of the product of negated log()'s in R. The user is required to cast para into estim

brute

Should brute force be used instead of two nested integrate() functions in Rto perform the double integration;

delta

The $\mathrm{d}w$ for the brute force (brute=TRUE) integration; and

...

Additional arguments to pass.

Value

• The value for $\rho_E$ is returned.

encoding

utf8

References

Joe, H., 2014, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p.

See Also

rhoCOP, tauCOP

Examples

Theta <- GHcop(tau=1/3)$para     # Gumbel-Hougaard copula with Kendall's Tau = 1/3
rhobevCOP(cop=GHcop, para=Theta) # 0.3689268 (RhoE after Joe [2014])
rhoCOP(   cop=GHcop, para=Theta) # 0.4766613 (Spearman's Rho)
set.seed(394)
simUV <- simCOP(n=30000, cop=GHcop, para=Theta, graphics=FALSE) # large simulation
samUV <- simUV * 150; n <- length(samUV[,1]) # convert to fake unit system
samUV[,1] <- rank(simUV[,1]-0.5)/n; samUV[,2] <- rank(simUV[,2]-0.5)/n # hazen
rhobevCOP(para=samUV, as.sample=TRUE) # 0.3708275