CopulaCenR (version 1.2.3)

tau_copula: Calculate Kendall's tau

Description

To obtain Kendall's tau from copula parameter(s)

Usage

tau_copula(eta, copula)

Value

Kendall's \(\tau\)

Arguments

eta

copula parameter(s); if copula = "Coupla2", input \(\alpha\) and \(\kappa\)

copula

specify the type of copula model

Details

The supported copula models are "Clayton", "Gumbel", "Frank", "AMH", "Joe" and "Copula2". The "Copula2" model is a two-parameter copula model that incorporates Clayton and Gumbel as special cases.

The Kendall's \(\tau\) formulas are list below:

The Clayton copula Kendall's \(\tau = \eta/(2+\eta)\).

The Gumbel copula Kendall's \(\tau = 1 - 1/\eta\).

The Frank copula Kendall's \(\tau = 1+4\{D_1(\eta)-1\}/\eta\), in which \(D_1(\eta) = \frac{1}{\eta} \int_{0}^{\eta} \frac{t}{e^t-1}dt\).

The AMH copula Kendall's \(\tau = 1-2\{(1-\eta)^2 \log (1-\eta) + \eta\}/(3\eta^2)\).

The Joe copula Kendall's \(\tau = 1 - 4 \sum_{k=1}^{\infty} \frac{1}{k(\eta k+2)\{\eta(k-1)+2\}}\).

The Two-parameter copula (Copula2) Kendall's \(\tau = 1-2\alpha\kappa/(2\kappa+1)\).

Examples

# fit a Copula2-Semiparametric model
data(AREDS)
copula2_sp <- ic_spTran_copula(data = AREDS, copula = "Copula2",
              l = 0, u = 15, m = 3, r = 3,
              var_list = c("ENROLLAGE","rs2284665","SevScaleBL"))
tau_copula(eta = as.numeric(coef(copula2_sp)[c("alpha","kappa")]),
           copula = "Copula2")