# tauAMH: Ali-Mikhail-Haq ("AMH")'s and Joe's Kendall's Tau In copula: Multivariate Dependence with Copulas

## 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 - 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.

## Usage

 ```1 2``` ```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}.

## Details

`tauAMH()`:

For small `theta` (), 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`.

## Value

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.

## Examples

 ```1 2 3 4 5 6 7``` ```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()!) ```

### Example output

```[1] 0.000000e+00 2.021099e-13 2.119277e-07
```

copula documentation built on Nov. 17, 2017, 2:25 p.m.