rhobevCOP {copBasic}

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}\bigl[(-\log U) \times (-\log V)\bigr] - 1 = \int_0^1 \bigl[B(w)\bigr]^{-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\bigl[\mathbf{C}_{BEV}(\mathrm{exp}[-w], \mathrm{exp}[w-1])\bigr]\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 Rho

\rho_S = 12 \int_0^1 \bigl[1 + B(w)\bigr]^{-2}\,\mathrm{d}w - 3\mbox{,}

in terms of B(w). This expression, in conjunction with rhoCOP, was used to confirm the prior expression shown here for B(w) in terms of \mathbf{C}_{BEV}(u,v). Lastly, for independence (uv = \mathbf{\Pi}; P), \rho_E = 0 and for the Fréchet–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 R data.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 estimated probabilities (see Examples);

brute

Should brute force be used instead of two nested integrate() functions in R to 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.

Author(s)

W.H. Asquith

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 Tau = 1/3
rhobevCOP(cop=GHcop, para=Theta) # 0.3689268 (RhoE after Joe [2014])
rhoCOP(   cop=GHcop, para=Theta) # 0.4766613 (Spearman Rho)

## Not run: 
set.seed(394)
Theta <- GHcop(tau=1/3)$para     # Gumbel-Hougaard copula with Kendall Tau = 1/3
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
## End(Not run)