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 - 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

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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].

See Also

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

Examples

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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()!)

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