copula (version 0.999-15)

coeffG: Coefficients of Polynomial used for Gumbel Copula

Description

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.

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 $1/\theta$ in $(0,1]$; you may use mpfr(alph, precBits = ) for higher precision methods ("Rmpfr*") from package \href{https://CRAN.R-project.org/package=#1}{\pkg{#1}}RmpfrRmpfr.

method

a character string, one of

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 $$% latex a_k(\theta, d) = (-1)^{d-k}\sum_{j=k}^d \alpha^j * s(d,j) * S(j,k), k \in \{1,\ldots,d\}.$$

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)