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 $$\rho_E = \mathrm{E}\bigl[(-\log U) \times (-\log V)\bigr] - 1 = \int_0^1 \bigl[B(w)\bigr]^{-2}\,\mathrm{d}w - 1\mbox{,}$$ 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, ...)

Value

The value for $\rho_E$ is returned.

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.

Author

W.H. Asquith

References