Computes the absolute values of the dth generator derivative psi^{(d)} via Monte Carlo simulation.
1 2 3 
t 

family 
Archimedean family (name or object). 
theta 
parameter value. 
degree 
order d of the derivative. 
n.MC 
Monte Carlo sample size. 
method 
different methods:

log 
if TRUE the logarithm of absdPsi is returned. 
is.log.t 
if TRUE the argument 
The absolute value of the dth derivative of the LaplaceStieltjes transform psi=LS[F] can be approximated via
(1)^d psi^{(d)}(t) = int_0^Inf x^d exp(tx) dF(x) ~= (1/N) sum(k=1..N)V_k^d exp(V_k t), t > 0,
where V_k ~ F, k in {1,...,N}.
This approximation is used where d=degree
and
N=n.MC
. Note that this is comparably fast even if
t
contains many evaluation points, since the random variates
V_k ~ F, k in {1,...,N} only have
to be generated once, not depending on t
.
numeric
vector of the same length as t
containing
the absolute values of the generator derivatives.
Hofert, M., Mächler, M., and McNeil, A. J. (2013). Archimedean Copulas in High Dimensions: Estimators and Numerical Challenges Motivated by Financial Applications. Journal de la Société Française de Statistique 154(1), 25–63.
acopulafamilies
.
1 2 3 4 5 6 7 8 9 10 11 12 13 14  t < c(0:100,Inf)
set.seed(1)
(ps < absdPsiMC(t, family="Gumbel", theta=2, degree=10, n.MC=10000, log=TRUE))
## Note: The absolute value of the derivative at 0 should be Inf for
## Gumbel, however, it is always finite for the Monte Carlo approximation
set.seed(1)
ps2 < absdPsiMC(log(t), family="Gumbel", theta=2, degree=10,
n.MC=10000, log=TRUE, is.log.t = TRUE)
stopifnot(all.equal(ps[1], ps2[1], tolerance=1e14))
## Now is there an advantage of using "is.log.t" ?
sapply(eval(formals(absdPsiMC)$method), function(MM)
absdPsiMC(780, family="Gumbel", method = MM,
theta=2, degree=10, n.MC=10000, log=TRUE, is.log.t = TRUE))
## not really better, yet...

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