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.

tauAMH():: Analytically, given as $$1-\frac{2((1-\theta)^2\log(1-\theta) + \theta)}{3\theta^2},$$ for theta$=\theta$; numerically, care has to be taken when $\theta \to 0$, avoiding accuracy loss already, for example, for $\theta$ as large as theta = 0.001.
tauJoe():: Analytically, $$1- 4\sum_{k=1}^\infty\frac{1}{k(\theta k+2)(\theta(k-1)+2)},$$ the infinite sum can be expressed by three $\psi()$ (psigamma) function terms.

Usage

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

Value

a vector of the same length as theta ($= \theta$), with

$\tau$ values

for tauAMH: in $[(5 - 8 log 2)/3, 1/3] ~= [-0.1817, 0.3333]$, of $\tau_A(\theta) = 1 - 2(\theta+(1-\theta)^2\log(1-\theta))/(3\theta^2)$, numerically accurately, to at least around 12 decimal digits.

for tauJoe: in [-1,1].

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 $10^{-5}$.

Details

tauAMH():

For small theta ($=\theta$), we use Taylor series approximations of up to order 7, $$\tau_A(\theta) = \frac{2}{9}\theta\Bigl(1 + \theta\Bigl(\frac 1 4 + \frac{\theta}{10}\Bigl(1 + \theta\Bigl(\frac 1 2 + \theta \frac 2 7\Bigr) \Bigr)\Bigr)\Bigr) + O(\theta^6),$$ where we found that dropping the last two terms (e.g., only using 5 terms from the $k=7$ term Taylor polynomial) is actually numerically advantageous.

tauJoe():

The "sum" method simply replaces the infinite sum by a finite sum (with noTerms terms. The more accurate or faster methods, use analytical summation formulas, using the digamma aka $\psi$ function, see, e.g., https://en.wikipedia.org/wiki/Digamma_function#Series_formula.

The smallest sensible $\theta$ value, i.e., th for which tauJoe(th) == -1 is easily determined via str(uniroot(function(th) tauJoe(th)-(-1), c(0.1, 0.3), tol = 1e-17), digits=12) to be 0.2387339899.

Examples

Run this code

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()!)