copula (version 0.999-15)

tauAMH: Ali-Mikhail-Haq ("AMH")'s and Joe's Kendall's Tau

Description

Compute Kendall's Tau of an Ali-Mikhail-Haq ("AMH") or Joe Archimedean copula with parameter theta. In both cases, analytical expressions are available, but need alternatives in some cases.

Usage

tauAMH(theta)
tauJoe(theta, method = c("hybrid", "digamma", "sum"), noTerms=446)

Arguments

theta

numeric vector with values in $[-1,1]$ for AMH, or $[0.238734, Inf)$ for Joe.

method

string specifying the method for tauJoe(). Use the default, unless for research about the method. Up to copula version 0.999-0, the only (implicit) method was "sum".

noTerms

the number of summation terms for the "sum" method; its default, 446 gives an absolute error smaller than $10^{-5}$.

Value

a vector of the same length as theta ($= \theta$), with $\tau$ valuesfor tauAMH: in $[(5 - 8 log 2)/3, 1/3] ~= [-0.1817, 0.3333]$, of $ tau.A(t) = 1 - 2*((1-t)*(1-t)*log(1-t) + t) / (3*t^2)$, numerically accurately, to at least around 12 decimal digits.for tauJoe: in [-1,1].

Details

See Also

acopula-families, and their class definition, "acopula". etau() for method-of-moments estimators based on Kendall's tau.

Examples

tauAMH(c(0, 2^-40, 2^-20))
curve(tauAMH,  0, 1)
curve(tauAMH, -1, 1)# negative taus as well
curve(tauAMH, 1e-12, 1, log="xy") # linear, tau ~= 2/9*theta in the limit

curve(tauJoe, 1,      10)
curve(tauJoe, 0.2387, 10)# negative taus (*not* valid for Joe: no 2-monotone psi()!)