rhobevCOP | R Documentation |
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
.
rhobevCOP(cop=NULL, para=NULL, as.sample=FALSE, brute=FALSE, delta=0.002, ...)
cop |
A bivariate extreme value copula function—the function |
para |
Vector of parameters or other data structure, if needed, to pass to the copula; |
as.sample |
A logical controlling whether an optional R |
brute |
Should brute force be used instead of two nested |
delta |
The |
... |
Additional arguments to pass. |
The value for \rho_E
is returned.
W.H. Asquith
Joe, H., 2014, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p.
rhoCOP
, tauCOP
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)
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