PPMiss.copulas {PPMiss}

Copula functions and the corresponding derivative limit.

Description

Implemented copulas and the corresponding derivative limit, as \theta \to a, where a is such that C_a(u,v)=uv. An estimate for \theta is obtained based on the copula function used. and the derivatives are used to obtain an estimate for K_1. The functions ‘frank’, ‘amh’, ‘fgm’ and ‘gauss’ are shortcuts for copula::frankCopula(), copula::amhCopula(), copula::fgmCopula() and copula::normalCopula() from package ‘copula’, respectively.

Usage

frank

dCtheta_frank(u, v)

amh

dCtheta_amh(u, v)

fgm

dCtheta_fgm(u, v)

gauss

dCtheta_gauss(u, v)

Arguments

u

a real number between 0 and 1.

v

a real number between 0 and 1.

Format

An object of class frankCopula of length 1.

An object of class amhCopula of length 1.

An object of class fgmCopula of length 1.

An object of class normalCopula of length 1.

Details

The constant K_1 is given by

K_1 = \int_0^1\int_0^1\frac{1}{l_0(u)l_n(v)}\lim_{\theta\rightarrow a}\frac{\partial C_{\theta}(u,v)}{\partial\theta}\,dudv,

where I=[0,1], l_m(x):= F_m'\big(F_m^{(-1)}(x)\big) and \{F_n\}_{n \geq 0} is a sequence of absolutely continuous distribution functions

Value

Archimedean copula objects of class ‘frankCopula’, ‘amhCopula’ or a Farlie-Gumbel-Morgenstern copula object of class ‘fgmCopula’ or an elliptical copula object of class ‘normalCopula’. For details, see archmCopula, fgmCopula and ellipCopula.

The derivative functions return the limit, as \theta \to 0, of the derivative with respect to \theta, corresponding to the copula functions.