copula: math-fun – R documentation

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

math-fun

Sinc, Zolotarev's, and Other Mathematical Utility Functions

Description

sinc(x) computes the sinc function s(x) = sin(x)/x for x != 0 and s(0) = 1, such that s() is continuous, also at x = 0.

A..Z(x, a) computes Zolotarev's function to the power 1-a.

Usage

sinc(x)
A..Z(x, alpha, I.alpha = 1 - alpha)

Arguments

`x`	`numeric` argument in [0,pi], typically a vector.
`alpha`	parameter in (0,1].
`I.alpha`	must be `= 1 - alpha`, maybe more accurately when `alpha` is very close to 1.

Details

For more details about Zolotarev's function, see, for example, Devroye (2009).

Value

A..Z(x,alpha) is A~Z(x,alpha), defined as

sin(alpha*x)^alpha * sin((1-alpha)*x)^(1-alpha) / sin(x), x in [0,pi],

where alpha in (0,1] is alpha.

References

Devroye, L. (2009) Random variate generation for exponentially and polynomially tilted stable distributions, ACM Transactions on Modeling and Computer Simulation 19, 18, 1–20.

Examples

curve(sinc, -15,25); abline(h=0,v=0, lty=2)
curve(A..Z(x, 0.25), xlim = c(-4,4),
      main = "Zolotarev's function A(x) ^ 1-alpha")

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