copula: coeffG – R documentation

Become an expert in R — Interactive courses, Cheat Sheets, certificates and more!

coeffG

Coefficients of Polynomial used for Gumbel Copula

Description

Compute the coefficients a[d,k](θ) 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.

Usage

coeffG(d, alpha,
       method = c("sort", "horner", "direct", "dsumSibuya",
                  paste("dsSib", eval(formals(dsumSibuya)$method), sep = ".")),
       log = FALSE, verbose = FALSE)

Arguments

`d`	number of coefficients, (the copula dimension), d >= 1.
`alpha`	parameter 1/θ in (0,1]; you may use `mpfr(alph, precBits = <n_prec>)` for higher precision methods (`"Rmpfr*"`) from package Rmpfr.
`method`	a `character` string, one of `"sort"`: compute coefficients via exp(log()) pulling out the maximum, and sort. `"horner"`: uses polynomial evaluation, our internal `polynEval()`. `"direct"`: brute force approach. `"dsSib.<FOO>"`: uses `dsumSibuya(..., method= "<FOO>")`.
`log`	logical determining if the logarithm (`log`) is to be returned.
`verbose`	logical indicating if some information should be shown, currently for `method == "sort"` only.

Value

a numeric vector of length d, of values

a_k(θ, d) = (-1)^(d-k) Sum(j=k..d; α^j * s(d,j) * S(j,k)), k in 1..d.

Note

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 methods and its defaults may change in the future, and so the exact implementation of coeffG() is still considered somewhat experimental.

Examples

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)

copula

Multivariate Dependence with Copulas

v1.0-1

GPL (>= 3) | file LICENCE

Authors

Marius Hofert [aut] (<https://orcid.org/0000-0001-8009-4665>), Ivan Kojadinovic [aut] (<https://orcid.org/0000-0002-2903-1543>), Martin Maechler [aut, cre] (<https://orcid.org/0000-0002-8685-9910>), Jun Yan [aut] (<https://orcid.org/0000-0003-4401-7296>), Johanna G. Nešlehová [ctb] (evTestK(), <https://orcid.org/0000-0001-9634-4796>), Rebecca Morger [ctb] (fitCopula.ml(): code for free mixCopula weight parameters)

Initial release

2020-12-07