coeffG | R Documentation |
Compute the coefficients a_{d,k}(\theta)
involved in the
generator (psi) derivatives and the copula density of Gumbel copulas.
For non-small dimensions d
, these are numerically challenging to
compute accurately.
coeffG(d, alpha,
method = c("sort", "horner", "direct", "dsumSibuya",
paste("dsSib", eval(formals(dsumSibuya)$method), sep = ".")),
log = FALSE, verbose = FALSE)
d |
number of coefficients, (the copula dimension), d >= 1. |
alpha |
parameter |
method |
a
|
log |
logical determining if the logarithm ( |
verbose |
logical indicating if some information should be shown,
currently for |
a numeric vector of length d
, of values
% latex
a_k(\theta, d) = (-1)^{d-k}\sum_{j=k}^d \alpha^j * s(d,j) * S(j,k),
k \in \{1,\ldots,d\}.
There are still known numerical problems (with non-"Rmpfr" methods; and
those are slow), e.g., for d=100,
alpha=0.8 and sign(s(n,k)) = (-1)^{n-k}
.
As a consequence, the method
s and its defaults may change in
the future, and so the exact implementation of coeffG()
is
still considered somewhat experimental.
a.k <- coeffG(16, 0.55)
plot(a.k, xlab = quote(k), ylab = quote(a[k]),
main = "coeffG(16, 0.55)", log = "y", type = "o", col = 2)
a.kH <- coeffG(16, 0.55, method = "horner")
stopifnot(all.equal(a.k, a.kH, tol = 1e-11))# 1.10e-13 (64-bit Lnx, nb-mm4)
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