Two-dimensional Clayton Copula

Computes the joint cumulative distribution function (CDF) and probability density function (PDF) of the two-dimensional Clayton copula.

Clayton_Copula(u, v, theta)

Clayton_Copula_pdf(u, v, theta)

Arguments

u

Numeric vector of values in \([0,1]\). First marginal (uniform).

v

Numeric vector of values in \([0,1]\). Second marginal (uniform).

theta

Positive numeric scalar. Dependence parameter \(\theta > 0\).

Value

• Clayton_Copula: Numeric vector of CDF values.

• Clayton_Copula_pdf: Numeric vector of PDF values.

Details

The joint distribution function of the two-dimensional Clayton copula is $$ C(u,v;\theta) = \left(u^{-\theta} + v^{-\theta} - 1\right)^{-1/\theta}, $$ where \(\theta > 0\).

The corresponding joint density is given by $$ c(u,v;\theta) = (\theta + 1) u^{-(\theta + 1)} v^{-(\theta + 1)} \left(u^{-\theta} + v^{-\theta} - 1\right)^{-\left(1/\theta + 2\right)}. $$

References

Nelsen, R. B. (2006). An Introduction to Copulas. Springer.

Examples

u <- c(0.2, 0.5, 0.8)
v <- c(0.3, 0.6, 0.9)

Clayton_Copula(u, v, theta = 2)
#> [1] 0.1687632 0.4160251 0.7459638
Clayton_Copula_pdf(u, v, theta = 2)
#> [1] 1.901324 1.384699 1.856575