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

Description

Compute the measure of association known as the Hoeffding Phi $\Phi_\mathbf{C}$ of a copula from independence ($uv = \mathbf{\Pi}$; P) according to Cherunbini et al. (2004, p. 164) by

$$\Phi_\mathbf{C} = 3 \sqrt{10\int\!\!\int_{\mathcal{I}^2} \bigl(\mathbf{C}(u,v) - uv\bigr)^2\,\mathrm{d}u\mathrm{d}v}\mbox{,}$$

and Nelsen (2006, p. 210) shows this as (and absolute value notation by Nelsen helps in generalization)

$$\Phi_\mathbf{C} = \biggl(90\int\!\!\int_{\mathcal{I}^2} |\mathbf{C}(u,v) - uv|^2\,\mathrm{d}u\mathrm{d}v\biggr)^{1/2}\mbox{,}$$

for which $\Phi^2_\mathbf{C}$ (the square of the quantity) is known as the dependence index. Gaißer et al. (2010, eq. 1) have $\Phi^2_\mathbf{C}$ as the Hoeffding Phi-Square, and their definition, when square-rooted, matches Nelsen's listing.

A generalization (Nelsen, 2006) to $L_p$ distances from independence ($uv = \mathbf{\Pi}$; P) through the LpCOP function is

$$L_p \equiv \Phi_\mathbf{C}(p) = \biggl(k(p)\int\!\!\int_{\mathcal{I}^2} |\mathbf{C}(u,v) - uv|^p\,\mathrm{d}u\mathrm{d}v\biggr)^{1/p}\mbox{,}$$

for a $p: 1 \le p \le \infty$ and where $k(p)$ is a normalization constant such that $\Phi_\mathbf{C}(p) = 1$ when the copula $\mathbf{C}$ is $\mathbf{M}$ (see M) or $\mathbf{W}$ (see W). The $k(p)$ (bivariate definition only) for other powers is given (Nelsen, 2006, exer. 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 all 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.

Reflection/Radial and Permutation Asymmetry---Asymmetric forms similar to the above distances exist. Joe (2014, p. 65) shows two measures of bivariate reflection asymmetry or radial asymmetry (term favored in copBasic) as the distance between $\mathbf{C}(u,v)$ and the survival copula $\hat{\mathbf{C}}(u,v)$ (surCOP) measured by

$$L_\infty^{\mathrm{radsym}} = \mathrm{sup}_{0\le u,v\le1}|\mathbf{C}(u,v) - \hat{\mathbf{C}}(u,v)|\mbox{,}$$

or its $L_p^{\mathrm{radsym}}$ counterpart

$$L_p^{\mathrm{radsym}} = \biggl[\int\!\!\int_{\mathcal{I}^2} |\mathbf{C}(u,v) - \hat{\mathbf{C}}(u,v)|^p\,\mathrm{d}u\mathrm{d}v\biggr]^{1/p}\,\mathrm{with}\, p \ge 1\mbox{,}$$

where $\hat{\mathbf{C}}(u,v) = u + v - 1 + \mathbf{C}(1-u, 1-v)$ and again $p: 1 \le p \le \infty$. Joe (2014) does not seem to discuss and normalization constants for these two radial asymmetry distances.

Joe (2014, p. 66) offers analogous measures of bivariate permutation asymmetry (isCOP.permsym) ($\mathbf{C}(u,v) \not= \mathbf{C}(v,u)$) defined as

$$L_\infty^{\mathrm{permsym}} = \mathrm{sup}_{0\le u,v\le1}|\mathbf{C}(u,v) - \hat{\mathbf{C}}(v,u)|\mbox{,}$$

or its $L_p^{\mathrm{permsym}}$ counterpart

$$L_p^{\mathrm{permsym}} = \biggl[\int\!\!\int_{\mathcal{I}^2} |\mathbf{C}(u,v) - \hat{\mathbf{C}}(v,u)|^p\,\mathrm{d}u\mathrm{d}v\biggr]^{1/p}\,\mathrm{with}\, p \ge 1\mbox{,}$$

where $p: 1 \le p \le \infty$. Again, Joe (2014) does not seem to discuss and normalization constants for these two permutation symmetry distances. Joe (2014, p. 65) states that the “simplest one-parameter bivariate copula families [and] most of the commonly used two-parameter bivariate copula families are permutation symmetric.” The $L_\infty^{\mathrm{permsym}}$ (or rather a similar form) is implemented by LzCOPpermsym and demonstration made in that documentation.

The asymmetrical $L_\infty$ and $L_p$ measures identified by Joe (2014, p. 66) are nonnegative with an upper bounds that depends on $p$. The bound dependence on $p$ is caused by the lack of normalization constant $k(p)$. In an earlier paragraph, Joe (2014) indicates an upper bounds of 1/3 for both (likely?) concerning $L_\infty^{\mathrm{radsym}}$ and $L_\infty^{\mathrm{permsym}}$. Discussion of this 1/3 or rather the integer 3 is made within LzCOPpermsym.

The numerical integrations for $L_p^{\mathrm{radsym}}$ and $L_p^{\mathrm{permsym}}$ can readily return zeros. Often inspection of the formula for the $\mathbf{C}(u,v)$ itself would be sufficient to judge whether symmetry exists and hence the distances are uniquely zero.

Joe (2014, p. 66) completes the asymmetry discussion with three definitions of skewness of combinations of random variables $U$ and $V$: Two definitions are in uvlmoms (for $U + V - 1$ and $U - V$) and two are for $V-U$ (nuskewCOP) and $U+V-1$ (nustarCOP).

Usage

hoefCOP(     cop=NULL, para=NULL, p=2, as.sample=FALSE,
                                       sample.as.prob=TRUE,
                                       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, ...)

Value

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

Arguments

cop: A copula function;
para: Vector of parameters or other data structure, if needed, to pass to the copula;
p: The value for $p$ as described above with a default to 2 to match the discussion of Nelsen (2006) and the Hoeffding Phi of Cherubini et al. (2004). Do not confuse $p$ with $d$ described in Note;
as.sample: A logical controlling whether an optional R data.frame in para is used to compute the $\hat{\Phi}_\mathbf{C}$ (see Note). If set to -1, then the message concerning CPU effort will be surpressed;
sample.as.prob: When as.sample triggered, what are the units incoming in para? If they are probabilities, the default is applicable. If they are not, then the columns are re-ranked and divided simply by $1/n$---more sophisticated empirical copula probabilities are not used (EMPIRcop);
brute: Should brute force be used instead of two nested integrate() functions in R to 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.

Author

W.H. Asquith

References

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

Gaißer, S., Ruppert, M., and Schmid, F., 2010, A multivariate version of Hoeffding's Phi-Square: Journal of Multivariate Analysis, v. 101, no. 10, pp. 2571--2586.

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