AMHcop: The Ali--Mikhail--Haq Copula

Description

The Ali--Mikhail--Haq copula (Joe, 2014, p. 168) is $$\mathbf{C}_{\Theta}(u,v) = \mathbf{AMH}(u,v) = \frac{uv}{1 - \Theta(1-u)(1-v)}\mbox{,}$$ where $\Theta \in [-1,+1)$, where the right boundary, $\Theta = 1$, can sometimes be considered valid according to M<U+00E4>chler (2014). The copula $\Theta \rightarrow 0$ becomes the independence copula ($\mathbf{\Pi}(u,v)$; P), and the parameter $\Theta$ is readily computed from a Kendall Tau (tauCOP) by $$\tau_\mathbf{C} = \frac{3\Theta - 2}{3\Theta} - \frac{2(1-\Theta)^2\log(1-\Theta)}{3\Theta^2}\mbox{,}$$ and Spearman Rho (rhoCOP) by $$\rho_\mathbf{C} = 12\biggl(\frac{\mathrm{dilog}(1-\Theta)\times(1+\Theta) - 2\log(1-\Theta) + 2\Theta\log(1-\Theta)-3\Theta}{\Theta^2}\biggl)\; -\; 3\mbox{,}$$ where $\mathrm{dilog}(x)$ is the dilogarithm function (the Spence function): $$\mathrm{dilog}(x) = \int_1^x \frac{\log(t)}{1-t}\, \mathrm{d}t\mbox{.}$$ The support of $\tau_\mathbf{C}$ is $[(5 - 8\log(2))/3, 1/3] $ $\approx$ $[-0.1817258, 0.3333333]$ and the $\rho_\mathbf{C}$ is $[33 - 48\log(2), 4\pi^2 - 39]$ $\approx$ $[-0.2710647, 0.4784176]$, which shows that this copula has a limited range of dependency.

Usage

AMHcop(u, v, para=NULL, tau=NULL, ...)

Arguments

Nonexceedance probability $u$ in the $X$ direction;

Nonexceedance probability $v$ in the $Y$ direction;

para

A vector (single element) of parameters---the $\Theta$ parameter of the copula. However, if a second parameter is present, it is treated as a logical to reverse the copula ($u + v - 1 + \mathbf{AMH}(1-u,1-v; \Theta)$);

tau

Optional Kendall's Tau; and

...

Additional arguments to pass.

Value

Value(s) for the copula are returned. Otherwise if tau is given, then the $\Theta$ is computed and a list having

para

The parameter $\Theta$, and

tau

Kendall's tau.

and if para=NULL and tau=NULL, then the values within u and v are used to compute Kendall Tau and then compute the parameter, and these are returned in the aforementioned list.

References

M<U+00E4>chler, Martin, 2014, Spearman<U+2019>s Rho for the AMH Copula---A Beautiful Formula: copula package vignette, accessed on April 7, 2018 at https://CRAN.R-project.org/package=copula under the vignette rhoAMH-dilog.pdf.

Pranesh, Kumar, 2010, Probability distributions and estimation of Ali--Mikhail--Haq copula: Applied Mathematical Sciences, v. 4, no. 14, p. 657--666.

Examples

Run this code

# NOT RUN {
t <- 0.9 # The Theta of the copula and we will compute Spearman Rho.
di <- integrate(function(t) log(t)/(1-t), lower=1, upper=(1-t))$value
A <- di*(1+t) - 2*log(1-t) + 2*t*log(1-t) - 3*t
rho <- 12*A/t^2 - 3    # 0.4070369
rhoCOP(AMHcop, para=t) # 0.4070369
sum(sapply(100:1,function(k) exp(log(3)+k*log(t)-2*lchoose(k+2,2))))
# 0.4070369 (see Note, very many tens of terms are needed) 
# }
# NOT RUN {
# }
# NOT RUN {
layout(matrix(1:2,byrow=TRUE)) # Note Kendall Tau is same on reversal.
UVn <- simCOP(3560, cop=AMHcop, para=c(-0.9, "FALSE" ), col=4)
mtext("Normal definition [default]") # '2nd' parameter could be skipped
UVr <- simCOP(3560, cop=AMHcop, para=c(-0.9, "TRUE"),   col=2)
mtext("Reversed definition")
AMHcop(UVn[,1], UVn[,2])$tau # Should be about -0.1663313 because
AMHcop(UVr[,1], UVr[,2])$tau # tauCOP(AMHcop, para=-0.9) is theoretically. 
# }