Gumbel copula constructors

Construct functions that compute either the log density or the CDF of the bivariate Gumbel copula, intended for use with dcopula.

Usage

cgumbel(theta)

Cgumbel(theta)

Arguments

theta

Dependence parameter (\(\theta >= 1\)).

Value

A function of two arguments (u, v) returning either the log copula density (cgumbel) or the copula CDF (Cgumbel).

Details

The Gumbel copula density

$$ c(u,v;\theta) = \frac{e^{-A}}{uv} \, (\tilde u \tilde v)^{\theta - 1} S^{1/\theta - 2} (A + \theta - 1), $$ where \(\tilde u = -\log u\), \(\tilde v = -\log v\), \(S = \tilde u^\theta + \tilde v^\theta\) and \(A = S^{1/\theta}\), so that \(e^{-A}\) is the copula CDF.

See also

cgaussian(), cclayton(), cfrank()

Examples

x <- c(0.5, 1); y <- c(0.2, 0.4)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dbeta(y, 2, 1, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pbeta(y, 2, 1)
dcopula(d1, d2, p1, p2, copula = cgumbel(1.5), log = TRUE)
#> [1] -1.751222 -1.135706

# CDF version (for discrete copulas)
Cgumbel(1.5)(0.5, 0.4)
#> [1] 0.2770518