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 by Spearman Rho (rhoCOP), through M<U+00E4>chler (2014), by $$\rho_\mathbf{C} = \sum_{k=1}^\infty \frac{3\Theta^k}{{k + 2 \choose 2}^2}\mbox{.}$$

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, rho=NULL, tau=NULL, fit=c("rho", "tau"), ...)

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 ($u + v - 1 + \mathbf{AMH}(1-u,1-v; \Theta)$);

rho

Optional Spearmans's Rho from which the parameter will be estimated and presence of rho trumps tau;

tau

Optional Kendall's Tau from which the parameter will be estimated;

fit

If para, rho, and tau are all NULL, the the u and v represent the sample. The measure of association by the fit declaration will be computed and the parameter estimated subsequently. The fit has not other utility than to trigger which measure of association is computed internally by the cor function in R; 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))$value
A <- di*(1+t) - 2*log(1-t) + 2*t*log(1-t) - 3*t # Nelsen (2007, p. 172)
rho <- 12*A/t^2 - 3    # 0.4070369
rhoCOP(AMHcop, para=t) # 0.4070369
sum(sapply(1:100,function(k) 3*t^k/choose(k+2,2)^2)) # Machler (2014)
# 0.4070369 (see Note, very many tens of terms are needed) 
# }
# NOT RUN {
# }
# NOT RUN {
layout(matrix(1:2,byrow=TRUE)) # Note Kendall Tau is same on reversal.
s <- 2; set.seed(s); nsim <- 10000
UVn <- simCOP(nsim, cop=AMHcop, para=c(0.9, "FALSE" ), col=4)
mtext("Normal definition [default]") # '2nd' parameter could be skipped
set.seed(s) # seed is used to keep Rho/Tau inside attainable limits for the example
UVr <- simCOP(nsim, cop=AMHcop, para=c(0.9, "TRUE"),   col=2)
mtext("Reversed definition")
AMHcop2(UVn[,1], UVn[,2], fit="rho")$rho # -0.2581653
AMHcop2(UVr[,1], UVr[,2], fit="rho")$rho # -0.2570689
rhoCOP(cop=AMHcop, para=-0.9)            # -0.2483124
AMHcop2(UVn[,1], UVn[,2], fit="tau")$tau # -0.1731904
AMHcop2(UVr[,1], UVr[,2], fit="tau")$tau # -0.1724820
tauCOP(cop=AMHcop, para=-0.9)            # -0.1663313 
# }