tau_copula {CopulaCenR}

Calculate Kendall's tau

Description

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

Usage

tau_copula(eta, copula)

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).

Value

Kendall's \tau

Source

Ali MM, Mikhail NN, Haq MS (1978). A Class of Bivariate Distributions Including the Bi- variate Logistic. Journal of Multivariate Analysis doi:10.1016/0047-259X(78)90063-5.
Clayton DG (1978). A Model for Association in Bivariate Life Tables and Application in Epidemiological Studies of Familial Tendency in Chronic Disease Incidence. Biometrika doi:10.2307/2335289.
Gumbel EJ (1960). Bivariate Exponential Distributions. Journal of the American Statistical Association doi:10.2307/2281591.
Joe H (1993). Parametric Families of Multivariate Distributions with Given Margins. Journal of Multivariate Analysis doi:10.1006/jmva.1993.1061.
Joe H (1997). Multivariate Models and Dependence Concepts. Chapman & Hall, London.
Frank MJ (1979). On the Simultaneous Associativity of F(x, y) and x + y - F(x, y). Aequationes Mathematicae.

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")