AMHcop: The Ali--Mikhail--Haq Copula

Description

The Ali--Mikhail--Haq copula (Nelsen, 2006, p. 92--93, 172) 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 Mächler (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 Mächler (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. The infinite summation is easier to work with than Nelsen (2006, p. 172) definition of $$\rho_\mathbf{C} = \frac{12(1+\Theta)}{\Theta^2}\mathrm{dilog}(1-\Theta)- \frac{24(1-\Theta)}{\Theta^2}\log(1-\Theta)- \frac{3(\Theta+12)}{\Theta}\mbox{,}$$ where the $\mathrm{dilog(x)}$ is the dilogarithm function defined by $$\mathrm{dilog}(x) = \int_1^x \frac{\log(t)}{1-t}\,\mathrm{d}t\mbox{.}$$ The integral version has more nuances with approaches toward $\Theta = 0$ and $\Theta = 1$ than the infinite sum version.

Usage

AMHcop(u, v, para=NULL, rho=NULL, tau=NULL, fit=c("rho", "tau"), ...)

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 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.

Arguments

u: Nonexceedance probability $u$ in the $X$ direction;
v: 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 Spearman Rho from which the parameter will be estimated and presence of rho trumps tau;
tau: Optional Kendall 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.

Author

W.H. Asquith

References

Joe, H., 2014, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p.

Mächler, Martin, 2014, Spearman'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.

Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.