hoefCOP: The Hoeffding's Phi of a Copula or Lp Distances (Independence, Radial Asymmetry, or Reflection Symmetry Forms)

Description

Compute the measure of association known as Hoeffding's Phi $\sigma_\mathbf{C}$ of a copula from independence ($uv$, P) according to Cherunbini et al. (2004, p. 164) by $$\phi_\mathbf{C} = 3 \sqrt{10\int\!\!\int_{\mathcal{I}^2} (\mathbf{C}(u,v) - uv)^2\,\mathrm{d}u\mathrm{d}v}\mbox{,}$$ and Nelsen (2006, p. 210) shows this as $$\Phi_\mathbf{C} = \biggl(90\int\!\!\int_{\mathcal{I}^2} |\mathbf{C}(u,v) - uv|^2\,\mathrm{d}u\mathrm{d}v\biggr)^{1/2}\mbox{,}$$ for which $\Phi^2_\mathbf{C}$ (the square of the quantity) is known as the dependence index. A generalization (Nelsen, 2006) to $L_p$ distances from independence through the LpCOP function is $$L_p \equiv \Phi(p)_\mathbf{C} = \biggl(k(p)\int\!\!\int_{\mathcal{I}^2} |\mathbf{C}(u,v) - uv|^p\,\mathrm{d}u\mathrm{d}v\biggr)^{1/p}\mbox{,}$$ for a $p: 1 \le p \le \infty$ and where $k(p)$ is a normalization constant such that $\Phi(p)_\mathbf{C} = 1$ when the copula $\mathbf{C}$ is $\mathbf{M}$ (see M) or $\mathbf{W}$ (see W). The $k(p)$ is given (Nelson, 2006, Exercise 5.44, p. 213) in terms of the complete gamma function $\Gamma(t)$ by $$k(p) = \frac{\Gamma(2p+3)}{2[\Gamma(p + 1)]^2}\mbox{,}$$ which is implemented by the hoefCOP function. It is important to realize that the $L_p$ distances are symmetric nonparametric measures of dependence (Nelsen, 2006, p. 210). These are symmetric because distance from independence is used as evident by $uv$ in the above definitions.

Asymmetric forms similar to the above distances exist. Joe (2015, p. 65) however shows two measures of bivariate reflection asymmetry or radial asymmetry (term favored in copBasic) as the distance between the copula and the survival copula surCOP $$L_\infty^{(\mathrm{radsym})} = \mathrm{sup}_{0\le u,v\le1}|\mathbf{C}(u,v) - \hat\mathbf{C}(u,v)|\mbox{,}$$ or its $L_p^{(\mathrm{radsym})}$ counterpart $$L_p^{(\mathrm{radsym})} = \biggl[\int\!\!\int_{\mathcal{I}^2} |\mathbf{C}(u,v) - \hat\mathbf{C}(u,v)|^p\,\mathrm{d}u\mathrm{d}v\biggr]^{1/p}\mbox{with\ } p \ge 1\mbox{.}$$ Joe (2015) does not seem to discuss and normalization constants for these alternative distances. Joe (2015, p. 66) offers analogous measures of bivariate permutation asymmetry ($\mathbf{C}(u,v) = \mathbf{C}(v,u)$) defined as $$L_\infty^{(\mathrm{permsym})} = \mathrm{sup}_{0\le u,v\le1}|\mathbf{C}(u,v) - \hat\mathbf{C}(u,v)|\mbox{,}$$ or its $L_p^{(\mathrm{permsym})}$ counterpart $$L_p^{(\mathrm{permsym})} = \biggl[\int\!\!\int_{\mathcal{I}^2} |\mathbf{C}(u,v) - \hat\mathbf{C}(u,v)|^p\,\mathrm{d}u\mathrm{d}v\biggr]^{1/p}\mbox{with\ } p \ge 1\mbox{.}$$ Again, Joe (2015) does not seem to discuss and normalization constants for these alternative distances. The two asymmetrical measures of Joe are nonnegative numbers and evidently unbounded whose potential upper bounds depends on $p$. Yet in a previous paragraph likely concerning $L_\infty^{(\mathrm{radsym})}$, Joe indicates and upper bounds of 1/3. The numerical integrations for $L_p^{(\mathrm{radsym})}$ and $L_p^{(\mathrm{permsym})}$ can readily return zeros and often inspection of the formula for the copula itself would be sufficient to judge whether symmetry exists and hence the distances are uniquely zero.

Usage

hoefCOP(cop=NULL, para=NULL, p=2, brute=FALSE, delta=0.002, ...)
LpCOP(cop=NULL, para=NULL, p=2, brute=FALSE, delta=0.002, ...)
LpCOPradsym(cop=NULL, para=NULL, p=2, brute=FALSE, delta=0.002, ...)
LpCOPpermsym(cop=NULL, para=NULL, p=2, brute=FALSE, delta=0.002, ...)

Arguments

cop

A copula function;

para

Vector of parameters or other data structure, if needed, to pass to the copula;

The value for $p$ as described above with a default to 2 to match the discussion of Nelsen (2006) and Hoeffding's Phi of Cherubini et al. (2004);

brute

Should brute force be used instead of two nested integrate() functions in Rto perform the double integration;

delta

The $\mathrm{d}u$ and $\mathrm{d}v$ for the brute force (brute=TRUE) integration; and

...

Additional arguments to pass.

Value

The value for $\Phi(p)_\mathbf{C}$ is returned.

References

Cherubini, U., Luciano, E., and Vecchiato, W., 2004, Copula methods in finance: Hoboken, NJ, Wiley, 293 p.

Joe, H., 2015, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p.

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

Examples

Run this code

hoefCOP(cop=PSP) # 0.4547656
rhoCOP(cop=PSP) == hoefCOP(cop=PSP, p=1) # TRUE
LpCOP(cop=PLACKETTcop, para=1.6, p=2.6)  # Fractional p
set.seed(938) # Phi(1.6; Plackett) = 0.1184489; L_1 = 0.1168737
UV <- simCOP(cop=PLACKETTcop, para=1.6, n=2000, ploton=FALSE, points=FALSE)
hoefCOP(cop=PLACKETTcop, para=1.6, p=200)  # Large p near internal limits
L_1 <- 4*max(abs(PLACKETTcop(UV$U, UV$V, para=1.6) - UV$U*UV$V)) # p is infty
# and finite n and arguably a sample-like statistic here, now on intuition try
# a more sample-like means
U <- runif(10000); V <- runif(10000)
L_2 <- 4*max(abs(EMPIRcop(U, V, para=UV) - U*V)) # 0.1410254 (not close enough)
para <- list(alpha=0.15, beta=0.90, kappa=0.06, gamma=0.96,
             cop1=GHcop, cop2=PLACKETTcop, para1=5.5, para2=0.07)
LpCOPradsym( cop=composite2COP, para=para)
LpCOPpermsym(cop=composite2COP, para=para)
"MOcop.formula" <- function(u,v, para=para, ...) {
   alpha <- para[1]; beta <- para[2]; return(min(v*u^(1-alpha), u*v^(1-beta)))
}
"MOcop" <- function(u,v, ...) { asCOP(u,v, f=MOcop.formula, ...) }
 LpCOPradsym(cop=MOcop, para=c(.8,.5))
 LpCOPpermsym(cop=MOcop, para=c(.8,.5))