tauAMH {copula} | R Documentation |
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
for
theta
; numerically, care has to be taken when
, avoiding accuracy loss already, for example, for
as large as
theta = 0.001
.tauJoe()
:-
Analytically,
the infinite sum can be expressed by three
(
psigamma
) function terms.
Usage
tauAMH(theta)
tauJoe(theta, method = c("hybrid", "digamma", "sum"), noTerms=446)
Arguments
theta |
numeric vector with values in |
method |
string specifying the method for |
noTerms |
the number of summation terms for the |
Details
tauAMH()
:-
For small
theta
(), we use Taylor series approximations of up to order 7,
where we found that dropping the last two terms (e.g., only using 5 terms from the
term Taylor polynomial) is actually numerically advantageous.
tauJoe()
:-
The
"sum"
method simply replaces the infinite sum by a finite sum (withnoTerms
terms. The more accurate or faster methods, use analytical summation formulas, using thedigamma
akafunction, see, e.g., https://en.wikipedia.org/wiki/Digamma_function#Series_formula.
The smallest sensible
value, i.e.,
th
for whichtauJoe(th) == -1
is easily determined viastr(uniroot(function(th) tauJoe(th)-(-1), c(0.1, 0.3), tol = 1e-17), digits=12)
to be0.2387339899
.
Value
a vector of the same length as theta
(), with
values
for tauAMH
: in ,
of
,
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
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()!)