AMHcop: The Ali-Mikhail-Haq Copula

In wasquith/copBasic: General Bivariate Copula Theory and Many Utility Functions

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"), ...)

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.

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.

Note

Mächler (2014) reports on accurate computation of \tau_\mathbf{C} and \rho_\mathbf{C} for this copula for conditions of \Theta \rightarrow 0 and in particular derives the following equation, which does not have \Theta in the denominator:

\rho_\mathbf{C} = \sum_{k=1}^{\infty} \frac{3\Theta^k}{{k+2 \choose 2}^2}\mbox{.}

The copula package provides a Taylor series expansion for \tau_\mathbf{C} for small \Theta in the copula::tauAMH(). This is demonstrated here between the implementation of \tau = 0 for parameter estimation in the copBasic package to that in the more sophisticated implementation in the copula package.

  copula::tauAMH(AMHcop(tau=0)$para) # theta = -2.313076e-07

It is seen that the numerical approaches yield quite similar results for small \tau_\mathbf{C}, and finally, a comparison to the \rho_\mathbf{C} is informative:

  rhoCOP(AMHcop, para=1E-9)    # 3.333333e-10 (two nested integrations)
  copula:::.rhoAmhCopula(1E-9) # 3.333333e-10 (cutoff based)
  theta <- seq(-1,1, by=.0001)
  RHOa <- sapply(theta, function(t) rhoCOP(AMHcop, para=t))
  RHOb <- sapply(theta, function(t) copula:::.rhoAmhCopula(t))
  plot(10^theta, RHOa-RHOb, type="l", col=2)

The plot shows that the apparent differences are less than 1 part in 100 million—The copBasic computation is radically slower though, but rhoCOP was designed for generality of copula family.

Author(s)

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.

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

See Also

P

Examples

## 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 section, very many tens of terms are needed) 
## End(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 used to keep Rho/Tau inside attainable limits
  UVr <- simCOP(nsim, cop=AMHcop, para=c(-0.9, "TRUE"),   col=2)
  mtext("Reversed definition")
  AMHcop(UVn[,1], UVn[,2], fit="rho")$rho # -0.2581653
  AMHcop(UVr[,1], UVr[,2], fit="rho")$rho # -0.2570689
  rhoCOP(cop=AMHcop, para=-0.9)           # -0.2483124
  AMHcop(UVn[,1], UVn[,2], fit="tau")$tau # -0.1731904
  AMHcop(UVr[,1], UVr[,2], fit="tau")$tau # -0.1724820
  tauCOP(cop=AMHcop, para=-0.9)           # -0.1663313 
## End(Not run)

wasquith/copBasic documentation built on Dec. 13, 2024, 6:39 p.m.