Afunction: the Pickands dependence function A in survival MGL-EV copula
In lizhengxiao/rMGLReg: MGL copula regressions

Description Usage Arguments Details Value Examples

the Pickands dependence function A in survival MGL-EV copula

1	Afunction(w, param)

`w`	w \in [0,1].
`param`	copula parameter in survival MGL-EV copula.

In the bivariate case d=2 the stable tail dependence function \ell can be represented in terms of the Pickands dependence function A:

\begin{align} A_{δ}≤ft(w\right)=w I_{{\frac{1}{2}, \frac{1}{δ}+\frac{1}{2}}}≤ft[\frac{≤ft(1-w \right)^{-δ}}{≤ft(1-w \right)^{-δ} + w^{-δ} }\right] + ≤ft(1-w\right) I_{{\frac{1}{2}, \frac{1}{δ}+\frac{1}{2}}}≤ft[\frac{w^{-δ}}{≤ft(1-w \right)^{-δ} + w^{-δ} }\right]. \end{align}

where A_{δ} is the Pickands dependence function of survival MGL-EV copula. Here I_{m,n}^{-1}() denotes the inverse of the beta cumulative distribution function I_{m,n}() (or regularized incomplete beta function) with parameters shape1 = m and shape2 = n implemented by R's qbeta and pbeta respectively.

The extreme value copula \bar{C}^{MGL-EV} of the survival MGL copula is given by

\bar{C}^{MGL-EV}(u_1,u_2;δ)=\exp≤ft[\log≤ft(u_1u_2\right)A_{δ}≤ft(\frac{\log≤ft(u_2\right)}{\log≤ft(u_1u_2\right)}\right)\right].

the value of the Pickands dependence function in [0.5,1]

Afunction(w = 0.7, param = 1.2)

delta <- 0.7
w.vector <- seq(0.0001, 0.9999, length.out = 100)
y.vector <- Afunction(w = w.vector, param = delta)
plot(w.vector, y.vector, lwd = 2, col = 3, ylim = c(0.5, 1))

delta <- 1
w.vector <- seq(0.0001, 0.9999, length.out = 100)
y.vector <- Afunction(w = w.vector, param = delta)
lines(w.vector, y.vector, lwd = 2, col = 4)

delta <- 3
w.vector <- seq(0.0001, 0.9999, length.out = 100)
y.vector <- Afunction(w = w.vector, param = delta)
lines(w.vector, y.vector, lwd = 2, col = 5)

delta <- 5
w.vector <- seq(0.0001, 0.9999, length.out = 100)
y.vector <- Afunction(w = w.vector, param = delta)
lines(w.vector, y.vector, lwd = 2, col = 1)