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 - 2((1-t)(1-t)log(1-t) + t)/(3*t^2),*for

`theta`

*=t*; numerically, care has to be taken when*t -> 0*, avoiding accuracy loss already, for example, for*t*as large as`theta = 0.001`

.`tauJoe()`

:-
Analytically,

*1- 4 sum{k=1:Inf; 1/(k(t*k+2)(t(k-1)+2))},*the infinite sum can be expressed by three

*ψ()*(`psigamma`

) function terms.

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`theta` |
numeric vector with values in |

`method` |
string specifying the method for |

`noTerms` |
the number of summation terms for the |

`tauAMH()`

:-
For small

`theta`

(*=θ*), we use Taylor series approximations of up to order 7,*tau[A](th) = 2*th/9 *(1 + th*(1/4 + th/10* (1 + th*(1/2 + th*2/7)))) + O(th^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*ψ*function, see, e.g., http://en.wikipedia.org/wiki/Digamma_function#Series_formula.The smallest sensible

*θ*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`

.

a vector of the same length as `theta`

(*= θ*), with
*τ* values

for `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].

`acopula-families`

, and their class definition,
`"acopula"`

. `etau()`

for
method-of-moments estimators based on Kendall's tau.

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